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AstroNuclPhysics ® Nuclear Physics - Astrophysics - Cosmology - Philosophy | Gravity, black holes and physics |
Chapter 1
GRAVITATION AND ITS PLACE IN PHYSICS
1.1. Development of knowledge about nature,
universe, gravity
1.2. Newton's
law of gravitation
1.3. Mechanical
LeSage hypothesis of the nature of gravity;
1.4. Analogy
between gravity and electrostatics
1.5. Electromagnetic
field. Maxwell's equations.
1.6. Four-dimensional spacetime and special theory of relativity
1.6. Four-dimensional spacetime and special theory of relativity
Spacetime and Relativity
We will assume
that the reader is familiar with the basics of special theory of relativity (STR), or has at least some
awareness of it *). For the coherence and compactness of the
book, however, we include a chapter with a brief explanation of the
special theory of relativity from a somewhat more general point of view, leading to the theory of space,
time, electromagnetism and especially gravity.
*) The basics of STR are now included in
the curriculum of high school physics. For a more detailed study
of STR, we can recommend a great monograph by V.Votruba [263],
then the relevant chapters in [183], [250], [271], [135], etc.
^{ }The aim of this chapter is not a
detailed and comprehensive interpretation of the special theory
of relativity, but rather recall and emphasize key
elements of logical structures of
relativistic physics and give an overview of the basic
concepts, phenomena and relations of the special theory of
relativity, to which we will refer in the following. We will also
get acquainted with the geometric properties of 4-dimensional
spacetime and give a 4-dimensional
tensor formulation of the laws of mechanics and electrodynamics;
this will often be used in subsequent chapters in terms of general theory of relativity, astrophysics, and cosmology .
Points and
events in space and time
From a factual point of view, nature can be considered a world of
events: every physical process can be divided into a sequence
of individual elementary events. The event is, for example, the
collision of two particles, the decay of the nucleus of an atom,
the flashing of a lamp. The movement of the test particle is a
sequence of events "the
particle occurs at a certain place in a certain time moment". Experience teaches us
that each event can be completely and unambiguously characterized
by four numbers: the place
"where" happened
(3 spatial coordinates x, y, z *) and the time "when" happened (time instant of t ).
*) Numerical values and geometric meaning of these coordinates of
individual points depend on the coordinate system
used. The most used is the orthonormal so-called Cartesian
coordinate system, which was introduced for the two-dimensional
case of the plane by René Descartes (1596-1650); the name "
Cartesian " is related to the Latin transcription
of Descartes' name "Cartesius". This system
consists of mutually perpendicular lines X, Y, Z - axes,
intersecting at a common point called the origin
of the coordinate system O (from Latin Origin ).
The position (x, y, z) of any point P(x, y, z) is given here by
the intersections of the perpendiculars running from this point
to the individual axes X, Y, Z; we assign zero values of
coordinates x = y = z = 0 to the origin O. Oriented line r
from the beginning O to the point P we
call the guide or the position vector
(radius-vector) of the point P (for vectors see below).
In the next explanation we will meet even with more general
coordinate systems - linear oblique coordinates
(see below the section "Lorentz transformations and
relativistic kinematics", Fig.1.5c) and especially with curvilinear
coordinates (almost the rest of the book, starting with §2.1),
where the position of points is determined by the intersections
of precisely defined curves. The most common examples of
curvilinear coordinates in a plane are polar coordinates
(r, j),
in space spherical coordinates (r, J , j) - see §3.2,
§3.4.
Coordinate
transformations. Scalars, vectors, tensors.
The choice of origin, orientation and scale of the coordinate
axes is completely arbitrary; it is usually motivated only by the
greatest possible simplicity of expressing the studied task. In
general, however, we encounter coordinate systems whose origins
are shifted relative to each other, the axes are rotated relative
to each other, the scales on the axes are different. We then need
to find the transfer - transformation -
relationships between the coordinates of the points and other
quantities, expressed in both systems S and S'.
The conversion of point coordinate values, expressed in different
coordinate systems, is performed using algebraic-geometric
relations, resulting from the analysis of positional relations
between coordinate axes. In the case of a simple shift
(translation) of the origins O, O' of the coordinate system O(x_{o}, y , z_{o}), the relations
between the coordinates x, y, z and x', y', z' are given simply
x' = x-x_{o}
, y' = y-y_{o}
, z' = z-z_{o}
. When rotating the Cartesian reference system by a
certain angle a around some axis, the transformation relations are given
by the sinuses and cosines of the angle of rotation, eg around
the Z axis: x '= x.cos a
+ y.sin a
, y' = -x.sin a + y.cos a , z '= z. This is
an example of linear coordinate transformations.
For a more compact notation of transformation relations, it is
advantageous to denote the coordinates not by different letters
(x, y, z), but by different indices: (x_{1} , x_{2} , x_{3}). The linear transformation can then be expressed by
the equation x'_{i} = _{j = 1}S^{ }^{3} a _{ij} .x _{j} , i = 1,2,3. We get
an even more compact notation by introducing the so-called Einstein
summation convention : if an index occurs twice in an
expression, it means summation over this index, without
listing the summation character "S" and the summation
bounds (j = 1,2,3 in three-dimensional space) . Thus, we write
the given transformation equation x'_{i} = a _{ij}.x ^{j} ; it is further customary to write summation indices
one at the bottom and one at the top (here it is only formal, it
has its significance for the so-called covariant and
contravariant components of vectors and tensors, see below
"Four-dimensional vectors and tensors"). The
transformation equation x'_{i} = a_{ij} .x ^{j }also describes general transformations between curvilinear
coordinates, however, the coefficients a _{ij} are not constant,
but are functions of place (coordinates) - see §2.4.
Mathematical and physical quantities can be classified according
to their behavior in coordinate transformation :
¨ A scalar is a quantity that does not
depend on the choice of the coordinate system - it is invariant
(unchanged) during coordinate transformation. The numerical value
of a scalar physical quantity can only depend on the choice of
the units used (hence the name: lat. Scala = scale, ladder).
¨ Vectors in classical physics are called
quantities which, in addition to their size, also have a certain direction
in space (lat. Vector = rider, carrier). It is written
in bold or an arrow above the character, it is represented with
an arrow in space; the perpendicular projections of its length
into the coordinate axes form the components of the
vector. Typical examples are the position vector (guide,
radius-vector) of a certain point r with
components (x_{1} , x_{2} , x_{3}), velocity v with components v_{i} º dx_{i} /dt, momentum p with components p_{i} º m.v _{i} , acceleration a with components a _{i} º dv_{i }/dt, force F with components F_{i} º dp_{i }/dt. Since the direction is associated with the choice
of the reference coordinate system, its components are
transformed in the same way as the coordinates. From this point
of view, we define the vector A as a
trio of quantities A_{1} , A_{2} , A_{3} (folder or components of vector),
which is transformed in the same way as the coordinates
when the coordinates x'_{i} = a_{ij} .x ^{j }are^{ }transformed : A'_{i} = a _{ij} .A^{j}. The length, or absolute value or size of the vector,
is the quantity A º |A| º (A _{i} .A^{i} )^{1/2} , which is a scalar.
¨ Tensors are called sets
of quantities that are transformed as products of coordinates
during coordinate transformations. The second-order tensor is
called the set of quantities T_{ ij} , which are transformed according to the law T'_{ij} = a _{ik} .a _{jl} .T^{ kl} during the
transformations of the coordinates x'_{i} = a_{ik} .x ^{k} (summed over k, l). Tensors are generalizations of
vectors and describe quantities that are formed by products of
vector components (such as angular momentum or quadrupole
moment). The name "tensor " comes from the
Latin word tensio = stress; they were first used to
describe deformations of bodies by the action of a force vector
on a vector-oriented planar element. Higher order tensors are
defined analogously.
4-dimensional spacetime
From a
mathematical point of view, each event can be displayed as a
point in fictious four-dimensional space , the so-called spacetime (time-space), on
the axes of which three spatial coordinates and time are plotted;
points (events) in space-time are
called worldpoints.
The motion of a particle then corresponds
to a certain line - the so-called world line - in this
four-dimensional space-time, whose points determine the
coordinates of the particle at individual time points (it can be
said that over time, the world point corresponding to a given particle
is continuously displace in space-time and describes a
certain line - world line). A uniformly rectilinear moving
particle corresponds to a straight line, the accelerated motion
is expressed by a curved wordlline, the world line of a
particle "standing" at rest with respect to a given
frame of reference is a straight line parallel to the time axis.
From the physical point of view, the worldline expresses the kinematic history of a particle, because each worldpoint
expresses the position of the particle at a certain point in
space and at a certain time. Because we
cannot imagine spacetime in its four-dimensional form, one or two
spatial dimensions are omitted for graphical drawing, thus
creating a spatio-temporal diagram of the
observed precess (Fig.1.6).
^{ }The introduction of
four-dimensional spacetime in classical mechanics is so far only
purely formal. It does not define metrics either, because the
spatial dimension and the temporal dimension are not related in
any way. It is the finding of deep connections between space
and time and establishing metrics in four-dimensional
space-time, that is the main succes of the special theory of
relativity.
Classical Newtonian Mechanics
Classical mechanics is based on three of Newton's
laws :
1. Law of inertia :
A body which is not acted upon by an external force remains at
rest or in uniform rectilinear motion, ie v
º d r
/ dt = const.
Note:^{ }This formulation
refers to an idealized material particle without
macroscopic dimensions and internal structure. For real
macroscopic bodies, from a phenomenological point of view, the
law of inertia for translational motion can be supplemented by
the possibility of inertial rotation : " The
body remains at rest or in uniform rectilinear or rotational
motion until it is forced to change this state by force
". The adjective "external" in force may no longer
apply here - see eg " pirouette effect"for a
figure skater who reduces her moment of inertia by fitting her
hands (by internal force action) , which leads to an increase in her rotational speed.
^{ }However, these external complex
circumstances do not need to be taken into account in a
fundamental physical analysis - in fact, they are
summarized by the laws of mechanics for the individual
particles of which the body is composed. The
rotational inertial motion of a body is formed by a uniform circular
motion of individual particles of the body around a
rotational axis, in which the centrifugal force is compensated by
the mechanical strength of the body material (essentially electrical bonding forces between atoms and
molecules). Internally, it is an uneven
motion of particles with centripetal acceleration caused by
internal forces according to 2. Newton's law; it does not change
size, but only the direction of speed. These movements, arising
from the co-production of the law of inertia with the law of
force and acceleration, fall into the field of solid
mechanics , but are also applied in hydrodynamics .
It is not necessary to introduce rotational motion
into the basic formulation of the law of inertia - in the spirit
of "Occam's razor". The real fundamental law of
inertia thus lies in the above basic formulation
1 for translational motion.
2. Law of motion (force and
acceleration) :
Acceleration of a
body is directly proportional to the force acting on it, i.e. F = m. a , where F is the applied force, a º dv / dt º d^{2 }r / dt^{2} is the acceleration, m is the (inertial) mass of the body.
3. Law of action and reaction :
In the interaction
of two bodies, the force exerted by the second body on the first,
of the same magnitude but in the opposite direction than the
force exerted by the first body on the second: F_{AB} = - F_{BA} .
From a
formal-mathematical point of view, the first law is a special
case of the second law (for F
= 0, a = 0, ie v = const.) . Nevertheless,
the law of inertia has a fundamental and independent physical
meaning, because the terms "velocity",
"acceleration", "calm", "linear
motion" appearing in Newton's laws, can be defined only when
is pre-determined the frame of reference, with
respect to which the
motion of bodies is investigated. Newton's laws of 2nd and 3rd apply only in the inertial frame of reference, given by the law of inertia 1 .
^{ }These three basic laws of
"terrestrial" mechanics are associated by Newton's
law of gravitation (§1.2 "Newton's
law of gravity") , which is the starting point of
the so-called "celestial" mechanics of the motion of
stars, planets and moons around them.
Reference
system. Position and time measurement.
Whenever we talk about movement, we always mean movement in relation to the frame of
reference. The reference
system *) means a system of spatial
coordinates
indicating the position of bodies in space and a clock used to determine time
intervals.
The simplest way to measure the positional coordinates and
distances of bodies in space is by applying sufficiently rigid
and accurate - standard, ideal - measuring rods. The most
common way to measure time is to use a periodic
process (regularly
recurring); the criterion of correctness is that the periodicity
of one process agrees with the periodicity of others procesces.
Factors that affect only some such processes (eg material
temperature) are "non-
universal",
have a disruptive effect and must be removed or corrected during
objective measurement. In the following, we will assume that all
spatial and temporal measurements are performed using standard (ideal) clocks and measuring rods, ie
such rods and clocks for which all non-universal disturbances are
removed or corrected (discussed further in
the section "Exact - ideal - measuring space and
time"). In contrast, factors affecting all periodic processes in the same way (running
of all clocks) and the lengths of all measuring rods - universal
influences -
cannot be corrected in any way and must be considered as
influencing the course of time itself and the properties of space
itself. In modern physics, we do not look at space and time as
metaphysical categories, but as an expression of the relationship
between objects and events.
*) The terms "reference system" and
"coordinate system" are often merged.
It can be said that :
reference
system |
= | system
of spatio- temporal coordinates |
+ | the
way in which these coordinates are assigned to the individual points |
. |
Between the reference system and coordinate system is roughly the difference similar as between a landscape with real landmarks, and its map with cartographic coordinates. The reference system is based on certain real bodies forming "support points"; with their help, imaginary lines are drawn and individual places are provided with numbers - a system of coordinates is created. It should be noted that in general :
coordinate transformation | Ü / Þ | transition to another frame of reference | . |
From a physical point of view, however, it is usually not necessary to distinguish the two concepts too much (an exception is, for example, the issue of gravitational energy - see §2.8).
Exact - ideal - measurement of
space and time
For accurate measurement of physical quantities, it is generally
necessary to use such methods, aids and devices that are
sensitive enough to the measured quantity and are not
affected by other interfering influences and
circumstances of measurement. If this is not the case, at least
an accurate correction for these disturbances
and distortions must be possible. For the measurement of space
and time in fundamental physics, especially in the theory of
relativity, standard idealized clocks and
measuring rods are introduced as models :
Ideal
clocks^{ }
are calibrated clocks whose speed (frequency of used periodic
events) is not affected by any non-universal influences
such as temperature or applied forces. Thus, a pendulum or
hourglass clock would be completely unusable here (whose running
speed is directly determined by gravity, it stops in a weightless
state); similarly, other mechanical clocks could be affected by
mechanical deformations of their components. The most suitable in
this respect are electronic oscillators,
their most accurate variant is the atomic clock
:
Atomic clock
The electronic basis of the atomic
clock is a crystal-controlled oscillator - a
small precision-cut piezoelectric quartz crystal tuned to a high
Gigahertz frequency corresponding to the oscillations of the
atoms used. The relative accuracy of this electro-mechanical
oscillator alone reaches up to about 10^{-7}
(absolutely sufficient for most applications). Another substantial increase in accuracy is achieved
here by sensitive electronic tuning of the oscillator by means of
a feedback loop with a resonant frequency of the type of atoms
used. The most common are cesium atoms with a
resonant frequency of 9.192631770 GHz . The
amplified signal from the crystal oscillator is connected to a
radio wave transmitter to which cesium atoms are exposed in the
chamber. If the frequency - resonance - of the
oscillator coincides with the frequency of transition between the
ground and excited levels of hyperfine splitting of energy levels
in the cesium electron shell (caused by the
interaction of nucleus spins and electrons),
the cesium atoms go into the excited condition.
By applying a magnetic field, these excited atoms are separated
and detected. Using the number of excited cesium atoms, the frequency
of the crystal oscillator is continuously tuned
electronically in the feedback so that it
constantly coincides with the resonant frequency of the
transitions of the cesium atoms - 9 192 631 770 Hz
*). The number of these oscillations then measures
time very accurately. The source of the
exact frequency here comes directly from the electron
shell of the atoms, which is stable and is not affected
by any common external influences. In the end, we get a "atomic
shell-controlled oscillator " of cesium, which
achieves a relative accuracy of 10^{-13}.
*) Based on the atomic clock, a new
definition of the second was introduced in 1967 - as a
time interval corresponding to 9,192,631,770 electromagnetic
periods. radiation generated during the transition between two
levels of the very fine structure of the ground state of the
cesium 133-Cs atom.
Today's atomic clocks are quite complex and quite
large laboratory equipment. However, with advances in technology,
they can be expected to be at least partially miniaturized and
compacted in the near future so that they can be used in
"field" and space probes.
Ideal
measuring rods^{ }
are length-calibrated scales whose length is not affected by any non-universal
influences such as temperature or applied forces. Ideal
measuring rods should therefore be made of a non-thermally
expandable material, sufficiently strong and rigid.
^{ }If the influence of non-universal factors
cannot be avoided (which is usually not
100% possible), a correction
must be made for these non-universal influences. In physical
practice, especially in the theory of relativity,
"clocks" and "rods" are usually not used
directly to measure times and lengths, but more complex methods
using electromagnetic radiation - optical and radar
methods . Rather, measuring rods and precision clocks are used to
calibrate these methods.
In practice, the
reference system is always realized by some material bodies. The
reference system can be laboratory walls, the Earth's surface,
the center of our Galaxy, the walls of the space rocket cabin,
etc. In principle, any reference systems can be used, although in
specific cases some of them may be more suitable for describing
certain events than others. It is clear that for studying the
motion of the planets is preferable reference system
connection with the sun than the system with some of Jupiter
months, or for monitoring the tennis ball is more suitable
reference system consisting tennis court system than about
associated with passing cars...
^{ }Newton's first law is then a
statement that there are so-called inertial frames of
reference, in which the law of inertia applies. It is clear
that any reference system S', which moves uniformly in a straight
line with respect to a given inertial system S ,
is also inertial; thus there are an infinite number of inertial
systems. On the contrary, systems that move with respect to the
inertial system with non-zero acceleration are not inertial - the
law of inertia does not apply in them. The inertial frame of
reference is idealization; in the general theory of relativity it
is shown that global inertial systems do
not exist, but it is
always possible to find a local inertial
system which in
a sufficiently limited spatial area has all the properties of a
real inertial frame of reference.
Galileo
transformation and relativity
Consider two inertial frames of reference S and S' with parallel oriented Cartesian
spatial coordinates x, y, z and x', y ', z' (Fig.1.5a) such that
the system S' moves with respect to the system S in
the direction of the X axis at speed V ;
for the origin t = 0 = t' countdown of time in both systems we
choose the moment when the origins O
and O' of
both systems coincided. If we measure position coordinates and
time intervals in both systems with the same standard bars and
clocks (which we will always assume in the next one), the
relationship between coordinates and times measured in the
non-dashed and dashed system will be the
so-called Galileo transformation :
x = x '+ Vt, y = y', z = z ', t = t' . | (1.64) |
In the more general case, when the inertial system S' moves with respect to S at the velocity V in the general direction, the Galileo transformation has a vector form
r = r' + V . t , t = t ' . | (1.64 ') |
Galileo's transformation (1.64) is an expression of common kinematic and geometric notions resulting from everyday experience. From Galileo's transformation follows the common additive law of velocity addition: if a body moves with velocity v' with respect to the system S', then in the system S its velocity is
v = v ' + V , | (1.65) |
that is, the velocity of
the body in the non-dashed system is increased by the velocity V of the dashed system with respect to the
non-dashed system (resp., both velocities are composed vectorially) .
^{ }Experience expressed in classical (Galileo
and Newton) mechanics teaches, that there is no absolute rest
or absolute velocity of uniform rectilinear motion. Galileo's
principle of relativity argues that the laws of mechanics are
the same for every inertial frame of reference - all inertial
systems are equivalent in terms of classical mechanics; no
internal mechanical experiment can determine how fast a given
inertial system moves. Galileo came to this conclusion by
observing that the mechanical processes on a ship floating at a
constant speed on a calm surface proceed as if the ship were at
rest, so that mechanical experiments cannot make find out, that
the ship is at rest or moving in a straight line.
^{ }Indeed, the laws of classical
mechanics are invariant with respect to Galileo's
transformations (1.64). E.g. 2.Newton law F = m. a º m.d^{2 }x / dt ^{2} = m.d^{2} ( x + V
.t) / dt^{2}
= m.d^{2 }x
/ dt^{2} = F
(if the external
force does not depend on the velocity of motion of the body, ie F = F') retains its shape and the
numerical value of the coefficient of proportionality m in
Galileo transformations between by two inertial systems, similar
to any displacements or rotations of the spatial coordinate axes.
The laws of conservation of energy and momentum are also
invariant to Galileo's transformation.
^{ }In formulating Newton's laws of
classical mechanics, the fulfillment of two (seemingly) obvious
assumptions is tacitly assumed :
a)^{ }Assumption of universal (absolute) time, according to which the time
intervals between events are independent of the choice of the
reference system.
b)^{ }The distances of the current
positions of the points (and thus also the dimensions of the
bodies) are absolute , ie independent of the choice
of the reference system with respect to which the positions of
these points are determined.
^{ }Both of these assumptions are
contained in Galileo's transformation equations (1.64). Newton introduced the notion of "absolute
space" and
inertia was considered an attempt by material bodies to preserve
the "state of motion" in this
absolute space. However, the concept of absolute space is empty
in classical mechanics, because due to the validity of Galileo's
principle of relativity, nor the most careful examination of any
mechanical phenomena can determine which body or reference
system is in the "absolute rest". With no mechanical experiment by
themselves can not distinguish two inertial system. If some
physical laws differed for different relative moving observers, it would
be possible based on these differences to
determine, which
objects are in the space in (absolute) stillness and which move.
^{ }It has long been thought that electromagnetic phenomena are such phenomena that make it
possible to distinguish different inertial systems (and thus to
distinguish absolute motion and rest). Galileo's principle of
relativity proved to be incompatible with the classical Maxwell's
electrodynamics. If we use Galileo's relations (1.64) to mutually
transform equivalent quantities in the systems S and S' , Maxwell's equations will have a
different form. Electromagnetic
phenomena would therefore carried out differently in different
inertial systems. Maxwell's equations are not invariant with
respect to Galileo transformations. It follows from the law of
velocity composition (1.65) that if the speed of light with
respect to some "basic" inertial system S is equal to c , then with respect to another inertial
system S', this velocity decreases or increases as
the light beam moves in the direction or against the direction of
movement of the dashed system with respect to the non-dashed
system. The speed of light would therefore be different in
different inertial systems.
^{ }According to this, Maxwell's
theory could only apply in one of an infinite number of inertial
systems; we could consider this significant system as an
"absolute frame of reference" in accordance with
Newton's conception. According to the ether hypothesis, such a
system is represented by a stationary light-carrying ether,
or it could be a system connected to the center of gravity of all
the matter in the universe.
^{ }Accurate measurements by
Michelson and Morley, who (with the intention of directly
experimentally confirming the existence of the ether, determining
the absolute frame of reference and determining the speed of
absolute motion of the Earth relative to it) between 1881 and
1904 measured the speed of light in and against the direction of
the Earth motion, showed that the speed
of light in
a vacuum it is the same in different inertial systems.
Einstein's
special theory of relativity
Negative result of experiments Michelson
and Morley, which was later repeatedly and accurately re-verified,
physicists initially tried to explain (or rather to reconcile it
with the classical physical opinions) introducing some artificial
assumptions and additional hypotheses. However, these hypotheses do
not stand in confrontation with the results of other experiments
and observations. E.g. the simplest of them
- the assumption
that the ether is "entrained" around the Earth by its
motion and is therefore locally at rest with it - is incompatible
with the observed aberration of light of the stars. Lorentz on
the basis of the study of the motion of electrons stated a contraction
hypothesis, according to which the length of each body moving at
velocity v shortens
in the direction of movement in the ratio Ö(1 -
v^{2}/c^{2}) compared to its rest length.
However, the introduction of additional ad
hoc hypotheses,
which replace one mystery with another, cannot be a satisfactory
explanation for any phenomenon.
^{ }A.Einstein took a new, completely
principled position on the contradiction between mechanics and
electromagnetism, unburdened by prejudices of mechanistic ideas.
Einstein realized that measuring the speed of light in any
inertial system gives the same result c @ 2,998
.10^{8} m/s, which is not at all contradictory,
but on the contrary in full accordance with the principle of
relativity valid in mechanics. In his epoch - making work "On the electrodynamics of moving bodies" [78] Einstein generalized Galileo's principle of relativity from
mechanics to all physical phenomena
:
Theorem 1.1 (Eistein's special principle of relativity) |
The laws of physics are the same for all inertial frames of reference . |
Thus, all inertial systems are completely equivalent for the description of all physical
processes; under the same physical conditions, physical phenomena
take place in the same way in each inertial system, regardless of
the speed of its movement. Every physical experiment turns out
the same whether we perform it in any inertial system. Einstein's
special principle of relativity is thus an
expression of the undetectability and non-existence of a universal (absolute) reference
system.
^{ }The special principle of
relativity is also a reflection of the unity
of physics,
the common material essence of nature. No electromagnetic
experiment can not be carried out without the use of physical
bodies governed by the laws of mechanics and vice versa,
every mechanical action involves an electromagnetic interaction
between the particles of material of moving bodies. It follows
from the validity of (Galileo's) principle of relativity in
mechanics, that electromagnetic and other phenomena
should also comply with the principle of relativity.
In Newtonian mechanics,
of course, the special principle of relativity is fulfilled. In
order for the special principle of relativity to apply to electromagnetic phenomena described by Maxwell's equations, the quantity c
(contained in Maxwell's equations either directly or through
vacuum permittivity) indicating the
speed of propagation of electromagnetic waves in vacuum, in all
inertial systems must have the same
value (from a general-physical
point of view the speed of light is discussed in §1.1, passage
"Speed of light").
The application of the special principle of relativity to
electrodynamics thus naturally explains the result of Michelson's
experiment.
^{ }However, in the axiomatic
construction of a general theory, which should be the basis of
all physics, the use of complex Maxwell's equations (describing a
specific field of electromagnetic phenomena) as a starting axiom
is disadvantageous. Einstein therefore took the knowledge of the
constant of the speed of light as a primary independent postulate
along with the special principle of relativity :
Theorem 1.2 (principle of constant speed of light) |
The speed of light in vacuum is the same in all inertial systems regardless of any movement of the source or observer . |
Classical Newtonian physics is based on the assumption of the instantaneous interaction of bodies: a change the position (or generally some characteristic) of one of the interacting bodies will be reflected on the other bodies at the same time, regardless of their distance. Formally, this is expressed by describing the interaction of particles using the potential energy U(x_{1} , x_{2} , ..., x_{n} ), which is a function of only the positional coordinates of the particles x_{ i}. In reality, however, there is no direct, immediate action "in distance" in nature. If there is a change with one body, then on the other body that interacts with it, this change begins to manifest itself only after the certain final time interval. This time is needed for the interaction (the field that mediates it) to overcome the distance between the bodies. Thus, the interaction propagates at a finite rate, so that there is a certain maximum (limiting) rate of propagation of the interactions. From the first postulate of STR (special principle of relativity) it follows that this speed of propagation of interactions is the same in all inertial systems - it is therefore a universal constant. It follows from electrodynamics that this speed is equal to the speed of electromagnetic waves - the speed of light in vacuum c . The second basic postulate of STR can therefore also be formulated in the form :
Theorem 1.2 ' (principle of universal velocity of propagation of interactions) |
There is a maximum velocity of interactions propagation in a vacuum, equal to the speed of light c , which is the same for all inertial reference frames . |
Postulates 1.2 and 1.2' are not completely
equivalent;
formulation 1.2' excludes, for example, the possibility of
the existence of tachyons (particles moving at super-light
speed), because they could be used to interact at speeds in
excess of the maximum rate of propagation of the interactions.
However, tachyons can still be considered a figment of
physicists' imagination, as there are no theoretical or
experimental reasons for their existence (see below). The
problems of relativistic astrophysics and cosmology are not
affected by subtle differences in both formulations of the second
basic postulate of STR.
^{ }The second postulate of the special
theory of relativity - the existence of universal velocity, which
does not add up in size^{ }with no other speed - it is in sharp
contrast to the usual kinematic notions expressed by Galileo's
transformations and based on the concept of absolute space and
time. The usual rule of velocity addition does not apply here,
simple Galileo transformations of coordinates between inertial
systems must be replaced by more general transformations
(Lorentz). Spatial distance and time interval cease to be
objective absolute quantities *) , but depend on the reference
frame of which measures - become relative. The principles of
STR thus break down the usual intuitive concepts of space and
time, based on experience with conventional movement of
macroscopic bodies.
*) If the speed of light appears to be the
same for observers moving at different speeds, this is only
possible if their "watches" and "rulers" are
different - time and space are different for different observers.
It can be said that when the speed of light c
turned out to be absolute, the spatial scales
and time intervals must be relative ...
Who is right? - or is
wrong?^{ }
Within STR, observers moving at different speeds often differ
on the size of the length proportions, the duration of time
intervals, the time sequence (or present) of events. But it
doesn't have to be that one of them was right and the other was
wrong - everyone is right in its own frame of
reference... But what all observers must
legitimately agree on, is the objective existence
and course of natural processes ! Whether two
moving bodies collide or miss does, not depend on the observation
frame; from the point of view of different systems, only the time
indication may differ when and in what spatial coordinates this
happens..?.. These questions are further discussed below in the
passage "Paradoxes in the special
theory of relativity".
Lorentz transformations and relativistic kinematics
Together with
some other assumptions, such as the homogeneity
and isotropy of space and time and their Euclidean
geometric and topological properties, these two basic postulates 1.1 and 1.2
allow to establish new
transformation relations, generalizing Galileo's transformation (1.64) for the
transition from one inertial system to another, and to build a new kinematics and dynamics of the motion of material bodies - Einstein's special theory of relativity
(STR).
Terminological note:
The name " special"
because it is limited to inertial (evenly moving) systems, "relativity"
because only relative motion is physically important. The word
"relativity" further reflects the STR
conclusions that some physical quantities - such as temporal and
spatial intervals, present and same place of events, the weight
of body - lose their former absolute importance and become
relative magnitudes dependent moving reference frames
("observers"). However, one cannot agree with
the oft-cited statement that "according to the theory of
relativity, everything is relative"!
Some important quantities, such as the speed of light or
space-time intervals, on the other hand, are "absolute
", independent of the reference system (on the speed of
motion of the observer).
Consider an inertial system S with the origin of O , coordinates x, y, z and time t , and further inertial system S' with the origin O', the coordinates x', y', z' and the time t' which moves relative to the S with velocity V. The physical measurements in the reference system S 'are carried out in the same way (using the same aids - standard measuring rods and synchronized clocks) as in the system S' . At time t = t' = 0, let a light flash be sent from the beginning of O , which at this moment coincides with O' (Fig.1.5b). In the S system, the propagation of this light signal is expressed by the equation
s ^{2} º x ^{2} + y ^{2} + z ^{2} - c ^{2}.t ^{2} = 0 | (1.66) |
describing a spherical wavefront whose radius r = c.t increases with velocity c . In the reference system S' the light source moves at a speed -V, but due to the principle of constant speed of light (c' = c regardless of the speed and direction of movement of the source) the light signal propagation will look the same as in the system S :
(s') ^{2} º (x') ^{2} + (y') ^{2} + (z') ^{2} - (c.t') ^{2} = 0 . | (1.66 ') |
In order to comply with the principle of constant speed of light, it is necessary to assume different times in both systems. From the simultaneous fulfillment of equations (1.66) and (1.66') will emerge the sought transformation relations between the coordinates (t, x, y, z) and (t', x', y', z').
Fig.1.5. Coordinate transformations between inertial frames of
reference.
a) Galileo transformation. b) Derivation of Lorentz transformation. The light flash
emitted at time t = t' = 0
from the beginning O (which
at that time coincided with O' ) propagates on all sides at the same speed c
from the point of view of both systems S and S', so that at time t it fills the spherical wavefront o radius r =
ct, resp. r' = c.t'.
c) Geometric
representation of the Lorentz transformation. If the default
reference frame is S in
space-time ascribed (pseudo) Cartesian coordinate system ct, x,
then the transition to the moving frame of reference S'
geometrically means a deformation to oblique for the affine
sharp-angled coordinates c.t', x'.
Note: The
image of the clock symbolically shows the speed of time in the
systems S and S' (see below - time dilation).
According to the principle of relativity, a body moving uniformly in a straight line from the point of view of the system S must also move uniformly in a straight line in the system S' . Therefore, the coordinates x', y', z', t' must be linear functions of the coordinates x, y, z, t. In order for equations (1.66) and (1.66') to be satisfied simultaneously, s'^{2} = k.s^{2} must hold , where k is a factor. This coefficient cannot depend on coordinates and time, because different points and moments of time would not be equivalent, which contradicts the homogeneity of space and time. The coefficient k cannot depend on the direction of velocity V either, because we assume the space in STR is isotropic; k could be a function of at most the magnitude of the velocity V = |V|, i.e. s' ^{2} = k(V) .s^{ 2} . However, the systems S and S' are equivalent. Therefore, the same consideration made from the point of view of the system S' with respect to which the non-dashed system moves at the speed -V shows that s ^{2} = k (|-V|) .s'^{2} = k(V) .s'^{2}, from which it follows k^{2} = 1, so k = 1 (a positive sign applies in order to preserve the identity of the transformation of the system S to itself at V = 0). Quantity s , so-called space-time interval, defined in equations (1.66), thus remains invariant during the transformation between two inertial systems :
s'^{2} s x'^{2} + y'^{2} + z'^{2} - c^{2}.t'^{2} = x^{2} + y^{2} + z^{2} - c^{2}.t^{2} s s^{2} . | (1.67) |
Consider, as with the Galileo transformation, the special case of Fig.1.5, where the axes of the two reference frames are parallel and in the same sense, the axes X and X' coincide and the system S' moves with respect to S at a constant speed V in the positive X axis. Then if y = 0, y'= 0 must be at any z and similarly if z = 0, z' = 0 must be at any y (areas XY and X'Y', as well as areas XZ and X'Z', transform themselves). Therefore, y' = k.y, z' = k.z where the coefficient k the same reasons as before, with an interval dependent on x, y, z, may be the only function of V . Coefficient k there is (again due to the indistinguishability of both systems) equal to one, so the coordinates perpendicular to the direction of motion do not change: y' = y, z' = z. The special transformation sought will therefore have (due to linearity) the form
x' = A. x + B. t, y' = y, z' = z, t' = P. x + Q. t . | (1.68) |
Substituting into the invariant condition (1.67) we get
(A^{2}-c^{2}P^{2})x^{2} + 2(AB-c^{2}PQ)x.t + (B^{2}-c^{2}Q^{2}) = x^{2} - c^{2}.t^{2} .
This relationship must be fulfilled identically in all places of space and at all times, so that the coefficients at x and t on both sides must be equal to each other:
A^{2}-c^{2}P^{2} = 1 , AB-c^{2}PQ = 0 , c^{2}Q^{2} - B^{2} = c^{2} .
We obtain the fourth equation from it, of that system S' with respect to S moves along the axis X velocity V . The point O' at the moment t has the coordinates O' = (x = Vt, y = 0, z = 0) from the point of view of S , while from the point of view of S' there is still O' = (x' = 0, y' = 0, z' = 0). From the first equation (1.68) we get between A and B the relation x' = A.V.t + B.t = 0, ie A.V + B = 0. By solving this system of four equations we get the results for the transformation coefficients v (1.68),
A = 1/Ö(1-V^{2}/c^{2}) , B = -V/Ö(1-V^{2}/c^{2}) , P = (-V^{2}/c^{2})/Ö(1-V^{2}/c^{2}) , Q = 1/Ö(1-V^{2}/c^{2}) ,
wherein the negative sign for B and P and the positive sign for A and Q is again due to the identity of the transformation at V ® 0.
After substituting into (1.68) the special transformation is sought
(1.69) |
This transformation,
which generalizes the Galileo transformation (1.64) and
guarantees the fulfillment of both basic postulates of STR, is
called the Lorentz transformation. Even before the emergence of
the special theory of relativity, Lorentz and Poincaré showed
that Maxwell's equations of the electromagnetic field retain the
same shape in two mutually moving inertial systems S
and S' if between these systems not simple
Galilei transformations but more complex transformations (1.69), called
now Lorentz transformations. However, in his special theory
of relativity, A.Einstein gave a general derivation of these
transformations and showed that it is not just some peculiarity
of a particular (electromagnetic) field, but they control all fields and all motion - they are an
expression of the structural
properties of space and time.
Spatio-temporal
diagrams^{ }
For a clear graphical representation of spatial motions of bodies
as a function of time - their worldline - the so-called spae-time
diagrams in x,
t coordinates are often drawn. Since STR deals with motions close
to the speed of light, it is suitable in spacetime diagrams
instead of the simple time t to plot its c-multiple - the time coordinate x ° = ct, so that the scale on the time
axis is comparable to the scales on the spatial axes. Such a space-time
diagram on which the horizontal axis x and the time axis perpendicular
to it are marked (y
and z coordinates are omitted for simplicity), corresponding to the initial
reference system S , is shown in Fig.1.5c. On these
coordinate axes, the space-time coordinates of any world point
(event) in the reference system S can be read. In order to read
these events spacetime are also in the reference frame S' moving
relative to S in the direction of axis x with velocity V, the coordinate axes x' and x'° =
c.t' corresponding to the system S'
must be plotted on this diagram. The x' axis, which is given by the
condition t' = 0, is according to (1.69) the line c.t = (V/c)
.x; the axis t', given by the condition x' = 0, is the line x =
(V/c) .ct. Thus, as can be seen from Fig.1.5c, the transition to
another inertial system by means of Lorentz transformations
geometrically means the transition to an oblique system of space-time
coordinates, the axes of which are inclined with respect to the original
axes by an angle a given
by tg a = V/c. This angle of inclination a increases with the velocity of the system
S' relative to S;
at V ® c approaches 45°, where the x' and c.t' axes
coincide. From such a geometric expression of the Lorentz
transformation, the kinematic effects of STR, such as the contraction of lengths or the dilation
of time, follow
very clearly; the well-known paradox of
the clock is
also elegantly addressed here [232], [242] -
it is analyzed in more detail below in the passage "Paradoxes
STR" .
^{ }The inverted
Lorentz transformations from the system S' to S are obtained due to the equivalence
of both systems simply by exchanging the dashed and un dashed
coordinates in the relations (1.69) and replacing the velocity V by -V :
^{ }
(1.69 ') |
The general Lorentz transformation, valid at any direction of the velocity V of the inertial system S' with respect to the system S , can be obtained from the special Lorentz transformation (1.69 ) by first using auxiliary coordinates such that movement occurs along the X axis, is applied (1.69) and then perform a backward transformation to the original coordinates. The general Lorentz transformation is usually written in vector form
(1.69 '') |
where r º [O, (x, y, z)] is the position vector from the origin O to the event (t, x, y, z). Folding the two Lorentz transformations S ® S' and S ® S'' a proper Lorentz transformation between S and S'' only when the speed of system S'' to the system S' has the same direction as the velocity S' to the S . Physically, this is due to the fact that the magnitude of the speed of light c does not compose with any other speed, while the direction of the speed of light generally changes (aberrations of light, see below). Therefore, the general Lorentz transformation cannot be obtained by simply adding special Lorentz transformations in the individual X, Y, Z axes.
Kinematic
effects of STR
From Lorentz transformations (1.69) follow the known kinematic
effects of special
theory of relativity - time dilation,
length contraction and non-additive law of velocity addition :
^{ }If we have in the system S
two same-place events x, y, z, t and x, y, z , t + Dt separated by the time interval Dt, then according to (1.69), since Dx = 0 (same-place), the time interval between
these events measured from the system S'
will be equal to
D t ' = D t / Ö (1 - v ^{2} / c ^{2} ) . | (1.70) |
The time measured by an ideal clock moving with a given body is called the body's own time. This proper time t is related to the space-time interval by the relation (since dx = dy = dz = 0)
d t = (1 / c) . ds , | (1.71) |
and is therefore also invariant. From relation (1.70), or equivalent by introducing velocity v^{2} = (dx^{2}+dy^{2}+dz^{2})/dt^{2} in relation dt = ds/c = (1/c)Ö(-c^{2}dt^{2}+dx^{2}+dy^{2}+dz^{2}), we get
dt = Ö(1 - v^{2}/c^{2}) . dt ; | (1.72) |
the interval of the proper time of the moving body is therefore always smaller than the corresponding interval of the coordinate time. An observer comparing the movement of the rest and moving clocks finds that the moving clocks go according to the relation (1.70) the slower the faster they move; this phenomenon is called time dilation.
Fig.1.5 - presented for clarity again
a) In classical mechanics (Galileo transform) the velocity of
time is the same in all inertial systems, regardless of their
velocity.
b), c) In the special theory of relativity, the effect of time
dilation is applied - in a moving system S' time flows slower than in the initial rest system S
.
In pre-relativistic
physics, the simultaneity of two events taking place in
different places could be convinced by means of a suitable
signal, such as a light signal, for the speed of which the common
law of speed composition applied. Two current events in terms of
one frame of reference are then simultaneous in every other
inertial system - the concept of the present has absolute meaning in classical
physics and does not depend on the state of motion of the
observer. From the
Lorentz transformation (and actually has a simple consideration
of the independence of the speed of light on moving reference
frame) also suggests that the two events taking place in
different locations, which in terms of a reference system appears
to be present, runs in terms of other systems in different moments of
time. So it is in STR the simultaneity concept of relative, depends
on the state of motion of the observer. According to the STR, it
is necessary to use light signals to define the simultaneity,
for which the independence of their speed on the reference system
is guaranteed.
^{ }Similarly, the dimensions of
bodies and the distances between them in non-relativistic
kinematics do not depend on whether they are observed by a
resting or moving observer. To determine the length of a body
(rod, scale) in STR, it is necessary to determine the current
values of the coordinates x_{1} , y_{1} , z_{1} and x_{2} , y_{2} , z_{2 }of its ends
at a given moment in the given reference system S. The
length in the x direction is then Dx = x_{2}
- x _{1} , similarly in the y
and z
directions. If we do the same in terms of the reference system S' moving at
velocity V , then from the Lorentz
transformations (where Dt = 0 - present) it follows
D x = D x'/ Ö (1 -V^{2} / c^{2} ), D y = D y', D z = D z'. | (1.73) |
The proper length of a given rod means its length l_{ o} measured in the reference system with respect to which this rod is at rest. From relation (1.73) it follows that the length of the rod moving in the longitudinal direction at velocity v will be
l = l _{o} . Ö (1 - v^{2} / c^{2} ) . | (1.73 ') |
This finding, called the Lorentz contraction of lengths, says that the dimension of each body appears to be shortened in the direction of motion in the ratio Ö(1 - v^{2}/c^{2}) compared to the rest dimension; the dimensions perpendicular to the direction of movement do not change, they are the same as the rest ones.
The Lorentz transformation formulas (1.69) also show the relationships between the particle velocities measured in different inertial systems. If in a system S' moving relative to a system S with velocity V in the direction of the X axis , the investigated particle will have velocity v º (v'_{x} = dx'/ dt', v'_{y} = dy' / dt ', v'_{z} = dz'/ dt'), then from the relations (1.69) rewritten in differential form, flow for the components of the velocity v in the system S transformation relations
(1.74) |
representing Einstein's law of speed addition. In particular, if in S' a particle moves in the direction of the X- axis at velocity v , then the result of its composition with the velocity V (of the same direction) of the system S' relative to S will be
v = (v' + V) / (1 + v'.V / c^{2} ) . | (1.74 ') |
It can be seen that the
sum of two speeds less than or equal to the speed of light always
gives a speed not exceeding the speed of light. If in relation
(1.74) we set | v'| = c (maybe a photon), we get
|v| = Ö (v_{x }^{2} + v_{y }^{2} + v_{z }^{2}) = c - the speed of light does
not add up with any speed in size. Even if the system
S' with respect to S moved at a speed V = c and a
particle with speed V' = c in the direction of movement
of the system S' passed through the system S', the
resulting velocity of this particle with respect to S according to the relation (1.74 ') v = (c + c ) / (1 + cc / c ^{2}
) = c would still be it was again equal only to the speed of
light. This confirms the property of the speed of light c as
the upper limit of the possible speeds of movement.
If both velocities v i V are small compared to the speed of
light c , the formula (1.74) turns into the
common additive law of velocity^{ }composition (1.65),
ie v = v '+ V.
^{ }An important special case of
Einstein's law of velocity composition is the relation describing
the change in the direction of light propagation when moving
from one system to another inertial - so-called
aberration of light.
If the photon moves in the plane XY of the system S' so that the direction of its
movement relative to an axis X (i.e. the direction of the
velocity of motion V of the system S') makes
an angle J, the components of its velocity in the
system S' will be equal to v'_{x} = c.cos J', v'_{y}
= c.sin J'. For the angle J of motion of this photon in the
system S (v_{x} = c.cos J , v_{y}
= c.sin J ) follows from the transformation
relations (1.74)
sin J = [(1 - V^{2}/c^{2})/(1 + (V/c)cosJ')] sin J' , cos J = (cosJ' + V/c)/(1 + (V/c)cosJ') .
In the
case V « c (to
the first order to the members of the V/c), hence the angle of the
aberration of light DJ = J' - J classical relationship DJ = (V/c).sinJ .
Relativity
of the kinematic effects STR^{ }
It should be noted that the above-mentioned kinematic effects,
caused by the speed of movement of the body, are observable only relatively
when the observed body and the observer move relative to each
other. If the observer *) decided to "catch a moving object
in the act" (what the hell is there conjuring with those
scales and clocks..?!..), he would jump behind him,
catch up with him and start moving with him the same speed, he
would find nothing at all -
scales and clocks would be fine, and all relativistic effects
would disappear in such an observation..!..
There is no absolute point of view in the theory of
relativity .
*) The very word "observer
" must generally be taken with a "grain of salt"
in physics: he must be free from any subjective influences,
appearances and feelings! An objective "observer" can
also be an instrument or the course of a natural event ...
Paradoxes of the Special Theory of
Relativity
The unusual nature of kinematic
regularities of the special theory of relativity, which seemingly
contradict
"common sense" *), have raised (and
evokes in the lay public sometimes to this day) a number
of objections, often formulated using "paradoxes".
All these paradoxes are created by erroneous or inconsistent
application of STR laws (most often
forgetting the relativity of the present); part of the reasoning is done
relativistically, part classically: Þ
contradiction.
Now apparent paradoxes of this kind are reliably solved [232], [242], they have only historical
significance, but they have played an important role in
formulating and refining STR thought processes.
*) STR is not a theory of "common
sense", but - whether we like it or against our minds - it
describes the properties of the real space-time
in which we live. We can say that this theory is a real victory
of an objective understanding over of the
so-called "common sense", based on the limited
experience of the everyday life of people in our terrestrial
conditions ...
Paradox of time^{ }
Strangest outlet relativity seems the effect of time
dilation - claims of different running speed
time in different reference frames, for different
observers. If we have the initial inertial frame of reference S
and the second system S ', which moves at high speed V relative
to S, the observer in S will see that the clock in S' is slower
compared to its "rest" clock, according to the
relativistic dilation of time. However, the observer in S 'can
rightly claim that his system is "quiescent" and,
conversely, the system S moves them at speed -V, so that the
clock in S, on the other hand, goes slower. Who is right? This
apparent discrepancy is often formulated as the clock
paradox, also called the twin paradox :
^{ }In an imaginary ("sci-fi")
experiment, imagine two observers, A
and B , who are twins.of the same age (they may
have an accurate, "ideal" watch on their hands).
Observer A stays here on Earth (we
will not consider here its gravity, rotation and circulation), while B gets on the rocket and flies off on an
interstellar space travel at a speed close to the speed of light.
If they are connected by radio signals, according to STR, Earth
observer A will see a slower passage of time on rocket B
.; astronaut B will in turn register the time dilation in
terrestrial base A. After returning in a
few years, the two brothers meet again and compare their
age and watch. Will they have the same physical age *)
and time on the watch? - or which of them will be
"older" or "younger"?
*) When asked whether the traveler will age
in accordance with the course of his standard "ideal"
clock, biochemistry answers in the affirmative: aging is the
result of biochemical processes at the molecular and atomic
levels, the speed of which corresponds to the physical course of
time measured by standardized clocks. From a philosophical point
of view may be a slight exaggeration to say that "we are
all kind of hours - and our faces are dials of years "
... (A.Eddington)
^{ }For the relativistic
analysis of this imaginary experiment by STR we to
primarily introduce a permanent inertial reference system S
with the beginning of Oin the launch point, associated
with the "rest" observer A ; it remains
unchanged throughout the experiment. The motion of astronaut B
can be divided into 5 stages:
I. Acceleration of motion after ignition of
rocket engines at point O , at the end of which the rocket
reaches speed V in the direction of the O-x axis, close to the speed of
light c.
II. Steady motion of V speeds after shutting
down rocket engines from Earth to the observation target (perhaps
a distant star).
III. Accelerated motion - after reaching the
observation target (distant stars) , the rocket engines are re-ignited to change the
direction of the probe's movement to the opposite, to Earth.
IV. Steady motion again at a relativistic speed
-V
, after turning off the rocket engines, towards the Earth.
V. Decelerated motion after turning on rocket
engines to brake from high speed V , to land on Earth.
^{ }In this idealized imaginary experiment,
for simplicity, we will assume that the rocket engines are very
powerful (and cosmonaut B very
resistant to overload, as well as his standard watch) , so the acceleration phase (I.), reverse maneuver
(III.) and braking (V.) will be very fast, with a negligibly
short duration with respect to stages II. and IV. uniform motion
relativistic velocity V . To move astronaut B then we can draw a
space-time diagram :
Spacetime diagram of interstellar
flight and return of cosmonaut B in the analysis
of the "twin paradox". The
motion of astronaut B is shown by the stronger
line OK´-L´-M´´-N´´-P, which has short curved
sections OK´, L´-M´´ and N´´-P, corresponding to
the acceleration and braking of the rocket, and long
straight sections corresponding to the inertial movement
back and forth. Several lines of the present between the
system S ´ of the departing rocket and the
changed system S ´´ of the returning rocket are
marked by oblique thinner lines - they have the opposite
inclination! Note:^{ }Own oblique coordinate axes of S ´ and S´´systems^{ }they are not drawn in the diagram, the picture would become confusing. |
The world line of the rest observer A is
the vertical line O-P along the time axis t in the rest
inertial system S ; the countdown for observer A is
on the vertical axis t (resp. c.t). The movement of the
"twin" -cosmonaut B is shown by the line
O-K´-L´-M´´-N´´-P, which is first inclined to the right
after the start (section O-L´), then after the return maneuver
it breaks to the left (section M''-L'') and finally lands on
Earth in wordpoint P . In the temporal analysis between
two mutually moving inertial systems, it is generally necessary
to use coordinate lines of the present in the
space-time diagram, which are inclined obliquely
at an angle given by the ratio V/c (cf. Fig.1.5) . In our case, it is an important "trick" to
read the time to realize that after the reverse direction of
movement in stage III. already in stage IV. it is another
inertial system that has the lines of the present inclined in the
opposite way than in stage II. - in the "-V"! Detailed
analysis yields the result that the sum of segments A-L + M-P
displays a shorter time interval than the
corresponding segment O-P observer A . Thus, astronaut B
returns to the common point P in a shorter
time - younger - than the time between the
resting observer.A . In our simplified case, where
astronaut B flew back and forth at speed V (and the
sections of the accelerated motions are negligibly short), the difference of the time intervals Dt_{A} and Dt_{B} of both observers
will correspond to the standard formula for time dilation (1.72)
:
D t_{B} = D t_{A }. Ö
(1 - V^{2} / c^{2} ) .
So if astronaut B went to the
nearest star, Proxima Centauri, 4.2 light-years away, at a rate
of, for example, 0.8^{ }c (approx. 240,000 km / s) there
and back, then, according to Earth observer A, he would
return in 14 years; by this time, Earth observer A would
grow old. However, astronaut B would only age 8.2 years of
his own time on this flight, so he would be 5.5 years younger
than his Earth brother A when he returned .
^{ }In the general case of
two standard clocks that fly apart at a learned
moment and meet again later, the time difference
will depend on the "histories" of their movements - the
velocities and directions of inertial movements and the dynamics
of non-inertial changes. Resp. on the symmetries
of both movements. If both observers move symmetrically
- the rockets will fly away in opposite directions at the same
speeds and accelerations and then return again with the same
movements, the relativistic dilatations are annulled
at the meeting point. In the second extreme case - complete asymmetry,
which corresponds to the case discussed here, the full
value of the relativistic dilation of time is
manifested.
Note:^{ }The
popularization literature sometimes argues that the general
theory of relativity (GTR) must be used to solve the twin
paradox, because the traveler's frame of reference is
non-inertial: that the time difference arises in the phase of
braking and reversing the second observer's motion. This
statement is misleading and unconvincing; the introduction of GTR
is just another alternative solution, which is unnecessarily more
complicated and does not bring new information unless there are
"real" gravitational fields, excited by the mass-energy
distribution. In fact, the twins' own paradox can be correctly
solved within the special theory of relativity itself
using three inertial frames of reference: one
rest system S of the first observer and two different
of the moving systems S ´ and S ´´ of the second
observer as he moves back and forth, as outlined above.
The length paradox^{ }
The second strange conclusion of the special theory of relativity
is the effect of the contraction of lengths in
different frames of reference - different lengths for different
observers. If we have the initial inertial reference frame S
and the second system S', which moves relative to S
at a large velocity V , the observer in S will see that the standard
bars are in S' compared to its "rest" rods
shorter, according to the relativistic contraction of lengths.
However, the observer in S' can rightly claim that his system is
"rest" and, conversely, the system S moves them
at speed -V , so that the rods in S are truncated. Again,
the question arises "who is right?". This apparent
discrepancy is generally referred to as the paradox of
lengths and is illustrated by various moving bodies - a
bar and a barn, a car and a garage, a plane and a hangar, a train
and a station or tunnel. The simplest formulation of the "rod
and barn paradox" consists in the following :
^{ }Let's build a simple building (shelter,
shed, barn) in an imaginary experiment of length L = 10
m, which has a door in the front and rear wall. This barn is
managed (and front and rear door opens or closes) the observer A
in the the rest default reference frame S. Furthermore,
there is a distant observer B , which in the direction of
the centers of these two doors throws at a relativistic speed eg V = 0.8c (approx. 240,000 km/s) a rod
of length l = 12 m and will move with it as an observer B´
in inertial system S´. What happens when a bar enters and
passes through a barn? - Does the rod "fit"
inside or not? From the viewpoint of the observer A
the bar appears to be shortened to a length l´ = 7.2 m by a
relativistic contraction according to the formula (1.73') and
therefore it should fit well into a 10 m barn. On the contrary,
the length of the barn appears to the observer B´
shortened to L´ = 6 m, so he will expect problems when passing
his 12 m bar through the barn. Moments of opening and closing the
front and rear doors can be used to assess whether or not a
flying bar can fit in a barn. If, from the point of view of
system S, observer A closes both doors when the rod
is completely inside (the end of the rod
has passed through the rear door), the
shortening of the rod will be demonstrated. From the perspective
of the observer B but however, it looks different: the
back door was closed when my bar had already hit the front door;
my rod was longer than observer's A barn. Their
disagreement lies in the timing of closing the door. In what is
meant by the present of two distant
events (although here only a few meters
away, the time data differs by only picoseconds). From this point of view, the relationship of the
present between the systems S and S´ needs to be
analyzed using the lines of the present, parallel to the
axis X´ on an oblique space-time diagram (somewhat similar to the above figure "Paradox
of time"; relative to the total marginality of the
problem we are not drawn special picture...) . According to the observer A (in the reference
frame S ) are at some point the two ends of the rod inside
the barn . From the perspective of an observer B' the
ends of the rod were never simultaneously inside the barn,
the rod is longer than the barn. From a formal point of view, both
observers are right, it is in a way an “optical
illusion”. From a physical point of view, only the
situation where both observers meet in a “braked” state
and from the point of view of a common frame of reference is
easily will find out if the bar will fit in the barn or not.
Everything else is just "STR folclore",
which may be nice and interesting, but may no longer have
anything to do with real natural reality..!..
^{ }Only the physical interactions of
particles and bodies are important. If they run at high
(relativistic) velocities, the effects of time dilation and
length contraction can be realistically applied. - see eg "High - energy collisions of heavier nuclei.
Quark-gluon plasma.",
where it can be seen in the picture at the bottom left that at
high energies the nuclei collide not as "balls" but
as" flat disks", due to the contraction of length ...
Seemingly superluminal speeds of movement ?
Kinematic velocity, defined formally as the rate of change of
coordinates of an object or phenomenon, can in some cases
exceed the speed of light in a vacuum. A simple example can be a light
trail - a spot of a beam from a rotating collimated source on
a screen at a great distance (such as the movement of a light
"piggy" in the distance when rotating a mirror
reflecting the sun's rays, or the light trail of a lighthouse at
a great distance) . However, such superluminal speed is not
the speed of the photons that make up the light trail. The light
trail in different places is formed each time by different
photons (which come from the direction
perpendicular to the apparent movement of the trail). There is no causal relationship between individual
"pigs" in different places, no information is
transferred between them. Therefore, the STR principles are not
violated in any way.
^{ }In quantum physics , superluminal
speeds can be virtually applied in connection with quantum
fluctuations and relations of uncertainity (see, for example, part "Quantum
Physics" in the monograph
"Nuclear Physics and Physics of Ionizing Radiation"). Locally superluminal
velocities are possible for virtual particles, which are
created and destroyed in the course of interactions; their speeds
fluctuate between moving slower and faster than light. However,
for real particles , which exist in the initial and final
states of the interaction, superluminal velocities do not arise.
These initial and resulting particles do not interact with each
other, and intermediate virtual particles are unable to transmit
information.
A consequence of the principled non-locality
of the quantum description of particles using wave functions is a
phenomenon called quantum entanglement. It consists in the
fact that two particles, whose quantum state is originally
"linked" by a common wave function, remain connected in
a certain sense - correlated, even over an arbitrarily
large distance. If the state of one of the entangled particles
changes, the state of the other particle will also change,
"immediately" - a kind of " teleportation
" of information occurs, transfer at infinite speed (passage "Quantum entanglement and
teleportation" in the same
chapter) . However, for a real measurement
of the quantum state, information about the measurement in the
initial system is required, which must be transmitted through a
classical communication channel, i.e. at sublight speed.
Quantum teleportation therefore does not allow the transfer of
information at a speed greater than the speed of light.
^{ }Thus, there are no
superluminal processes in quantum mechanics or quantum
field theory that can be used to transfer matter or communicate
information at superluminal speeds.
^{ }In relativistic cosmology , in the
globally curved spacetime of an expanding universe, very distant
objects can move away from each other at speeds exceeding the
speed of light. In §5.3 and §5.4 "The Standard Cosmological Model.
Big Bang. Formation of the Structure of the Universe." it will be shown that galaxies distant more than
c/H (Hubble's constant H~ 70 km/s/Mpc) are moving away from each other at speeds greater than
lights c . They will disappear beyond the event horizon
and will no longer be able to interact with each other. In
relativistic cosmology, space itself expands along with
matter - or. expansion is a dynamic property of "free"
spacetime itself. Particles of matter are only carried along by
it. From this point of view, this is not a mechanical movement,
so the relative velocities of particles during cosmological
expansion can be superluminal (without violating the laws
of the special theory of relativity) . ...
^{ }Observation
of superluminal velocities of distant astronomical sources
If we observe a distant object (emitting light) at a distabce d,
moving through space with a real speed v in a certain
direction q, it will generally have some velocity component v.cos q towards us and
also perpendicular to our line of sight v.sin q. The
astronomically observed velocity v´ of shining objects is
determined in a straightforward way by multiplying the distance d
by the observed angular velocity w: v´ = d.w. The angular
velocity is determined from the change in the observed angular
position j_{1}, j_{2} of the object at two different times t_{1}, t_{2 }: w = ( j_{2} - j_{1})/(t_{2} -t_{1}). However, this will not
be the real speed v of the object, as we would measure at
the source, but only the transverse apparent speed v´,
created by the projection of the real speed in the direction of
the observer. This apparent transverse speed is the only
speed of astronomical objects that can be directly measured in
the sky. At the same time, the final speed of light is not
considered here, the instantaneous propagation is assumed. In
reality, however, for the transverse movement the time interval t_{2}-t_{1} between the two
observations is shortened due to the approach of the object to
the observer ...
^{ }From a relativistic point of view, this
phenomenon has interesting consequences at high velocities close
to the speed of light, which are often found in jets of
high-energy particles from accretion disks in quasars and active
galactic nuclei (in figure left).
^{ }If there is any astronomically observable
distinct structure in the jet, we can determine the speed of
movement of this structure in the sky by precisely observing its
position A, B at two times t_{1}, t_{2} (a few months apart) - in principle, we can measure the speed of
the gas in the jet. Detailed measurements of the light (and electromagnetic radiation in general) of jets from quasars and active galactic nuclei have
shown that in many cases the movement of the glowing gas
structures in the jets appears to be faster than light.
This deceptive effect occurs when the velocity of the jet is
close to the speed of light (>0.9c) and there is a high
component of the velocity towards the Earth (small jet angle q approx.
10°-40°). In such a case, as the observed structure in the jet
moves towards the Earth, the distance d_{2} is shortened compared to d_{1} and thus the time
delay of detection in both positions A, B by the value (v/c).cosq. As if the
observed structure had managed to overcome the distance between
the two places A-->B earlier. This means that the
apparent observed velocity v´ appears larger
than the actual velocity v, by the ratio
1/[1-(v/c).cosq]. The observed speed v' is not limited by the
speed of light c because it is caused by retardation
effects.
Actual and
apparent velocity of quasar jets. ^{ }Left: Jet from the accretion disk of the central black hole, directed at an angle q with respect to the observer. ^{ }Right: Trigonometric analysis of the movement of investigated element in the jet and the observed light rays in two times t_{1}.and t_{2 } |
The figure on the right shows
the geometrical situation during the emission of a certain
monitored element in the jet from point A at time t_{1} and its arrival at
point B at time t_{2 }. Light rays are sent from these places and at these
times to the observer O - two astronomical observations of the
position j_{1}, j_{2} on Earth are made at times t_{1}' and t_{2}'.
^{ }From the trigonometric analysis according
to the figure on the right, it follows
that the actual cross-transverse speed of the observed object
will be v_{T}
= v.sin q. The observed apparent transverse velocity (angularly
measured) of the monitoted object then, due to the shortened
distance d_{2}
= d_{1} -
v.(t_{2}-t_{1}).cos q *), and thus the
shorter detection time interval t_{2}´- t_{1}´ = (t_{2}-t_{1})/[1-(v/c).cos q], will have value v´= v. sin q
/[1 - (v/c). cos q].
*) Because the angle j, under which the details of
jets near the quasar are observed from Earth, is very small (cosj~1), it does not
need to be considered in the difference between the distances d_{1}
and d_{2}. For distant quasars, the velocity is corrected
by a factor of d.(1+z), where d is the distance and z
is the cosmological redshift, to give the value of the velocity
that would be measured by an observer moving with the source
(i.e. at rest relative to the source).
^{ }If the observed
object is moving towards the observer, the observed velocity may
be overestimated compared
to the actual one. For certain combinations of true velocity v
and beam angle q, the apparent transverse velocity can be greater
than c. It is, for example, when v>0.9 and
the angle q is in the range of around 5°-40°. With the opposite
movement - in the direction apart of the observer (jet to the
opposite side), on the contrary, the observed speed would be underestimated.
And when the jet is headed straight for Earth (q=0), of course no
motion is apparent...
^{ }This situation
often occurs in jets from active galactic nuclei - from accretion
disks around supermassive black holes - quasars - is discussed in
§4.8, section "Thick accretion disks. Quasars." and "Mechanism of quasars and active
galactic nuclei".
Relativistic dynamics
So far we have
investigated only the purely kinematic laws of the special theory of
relativity. By applying relativistic kinematics
to the laws of dynamics, relativistic dynamics is created,
providing other remarkable effects.
^{ }Newton's equation of motion of the mass
point dp/dt = F , which is invariant with respect to the
Galileo transformation, must be modified (generalized)
so that it is
invariant with respect to the Lorentz transformation, while at
low speeds it passed into the original Newton's equation. To each material particle moving with respect to the
inertial system S with velocity v is
assigned a momentum vector p
p = ^{def. } m . v
proportional to speed v; the
proportionality coefficient m represents the inertial mass of the particle.
^{ }In order for Newton's equation of
motion and the law of conservation of momentum to be compatible
with relativistic kinematics, the mass m
will no longer
be a motion-independent constant as in classical mechanics, but
will be a universal function m = f(v) of the particle velocity v º |v| (the direction of velocity
cannot depend, with respect on the isotropy of space). It
follows from the principle of relativity that during the
transition to another frame of reference S', against which the
observed particle moves at velocity v',
wil be p' = m'. v', where m' = (v') is the same function of argument v', as function m = f(v) of
argument v (form-invariance). The form of this function f is unambiguously given by the
requirement that the law of conservation of momentum applies in
any inertial system. The easiest way to reach it by analyzing a
collision of two identical particles is accomplished in terms of
two different reference systems S
and S' using
a relativistic kinematics, i.e. Lorentz transformations (shape
function f can also be
obtained from the
requirement that the p acted as a vector in the Lorentz
transformation). Comes out the result f(v) =
f(0)/Ö(1 - v^{2}/c^{2}), so that the mass of the
particle moving at velocity v
is equal to
m = m _{o} / Ö (1 - v ^{2} / c ^{2}) , | (1.75) |
where m_{o} is the proper or rest mass of the particle identical to mass in
Newtonian mechanics. The moving body thus exhibits a higher
inertial mass, gives greater resistance to
further acceleration. In our daily lives, the velocities of
bodies are small, so we do not observe any change in mass. In the
microworld, however, particles often move at speeds close to the
speed of light, and the change in mass is no longer negligible. In accelerators are prepared high-energy particles, which have a mass
many times higher than their rest mass.
^{ }At v ® c
the mass m increases above all limits, which is a
dynamic obstacle preventing bodies with non-zero rest mass m_{o}
can reach the speed
of light v = c. However, there are also particles (quantum) with zero rest mass m_{o} = 0, such as photons or hypothetical gravitons. For these particles with m_{o} = 0, the momentum can remain finite even
when the speed of light is reached (relation (1.75) gives an
indefinite expression 0/0 at v = c). The velocity of particles
with zero rest mass even must always be exactly
equal to the speed of light c and their relativistic mass is
given by the amount of energy they transmit (this energy is directly proportional to the frequency
of the wave whose quantum is the given particle: E =
h.f). The momentum
of such a particle with zero rest mass must then be reported
separately - independently of its velocity
(which is identically equal to c).
^{ }The velocity, and thus the momentum of a free particle, is constant over time. When a
particle interacts with its surroundings, the speed of its motion
generally changes, whereby the measure of the force
acting being the change in the momentum of the particle per unit
time :
F = ^{def. } d p / dt . | (1.76) |
It is
advantageous to keep this definition of
force also in
relativistic mechanics, because (unlike the product of mass and
acceleration) it leads to the equivalence of the law of action
and reaction with the law of conservation of momentum. If the
force F, which is the cause of the change in the
momentum of a particle, is given as a function of place and time,
the relation (1.76) is the equation of motion of the particle.
Unlike Newtonian mechanics, the variability of m
means that the force and acceleration vectors may not
have the same direction.
^{ }The work A
performed by a force F with a given particle of mass m
is, as in Newtonian mechanics, defined as the product of the
applied force and the distance traveled by the particle during
this action:
dA = ^{def. }F . d r . | (1.77) |
If the force F acts on an otherwise free particle, it can be assumed that the delivered work is converted into the kinetic energy of the particle:
d E_{kin} = ^{def. } dA = F . d r .
If the particle of mass m moves with velocity v, after substituting z (1.76) and (1.75) we get
dE_{kin} = m (d v / dt) .d r + (dm / dt). v
. d r = m v .d v - v^{2} dm = = m_{o }v .d v / Ö(1 - v^{2} / c^{2}) ^{3} = c ^{2} dm. |
(1.78) |
Integration from 0 to v creates a relationship
E_{kin} = m_{o} c^{2} / Ö(1 - v^{2} / c^{2}) - m_{o} c^{2} = c^{2} (m - m_{o}) | (1.79) |
indicating the kinetic
energy of a particle with rest mass m_{o} moving velocity v , i.e. with inertial mass m.
At velocities v << c small in comparison with the speed of
light, this relation takes on the approximate shape E_{kin} » (1/2) .m_{o} v^{2} corresponding to the known
formula for kinetic energy in classical mechanics.
^{ }Equation (1.78) indicates that the
increase in the kinetic energy of a body is accompanied by a
proportional increase in its (inertial) mass m.
The analysis of mechanical processes, such as the perfectly
inelastic collision of two mass bodies, using relativistic
kinematics and the law of conservation of energy shows, that a
similar relationship of direct proportionality applies between
the supplied energy and the increase of rest mass of the body,
while the conserved total energy
E = m. c^{2} = m_{o} c^{2} / Ö(1 - v^{2} / c^{2}) = E_{o} + E_{kin} | (1.80) |
consists from kinetic energy
E_{kin} = (m - m_{o}). c^{2} | (1.80a) |
and resting energy
E_{o} = m_{o} . c^{2} . | (1.80b) |
Between the changes of mass and energy applies the universal Einstein's relation "equivalence of mass and energy"
D E = D m . c ^{2} | (1.80c) |
regardless of what causes the change in energy or mass. From (1.80) and the definition of momentum p = m.v follows (by excluding v ) an important general relationship between energy and momentum:
E ^{2} = p ^{2} c ^{2} + m_{o }^{2} c ^{4} . | (1.81) |
The relations (1.75) and
(1.78) - (1.81), which are a dynamic consequence of relativistic
kinematics, have been precisely
verified by
experiments in atomic physics, nuclear physics and elementary
particle physics; they have already become
an "engineering
part" of nuclear technology.
^{ }In non-relativistic physics, two
completely separate and isolated laws of conservation applied:
matter and energy. There was no universal relationship between
the (inertial) mass and energy of a body. In Einstein's theory of
relativity, however, the general relation E = m.c^{2}
holds, according to which the mass m
and energy E of each material object are mutually proportional
to the universal coefficient c^{2}. Mass and energy, which in
classical physics describe qualitatively different properties of matter, in relativity theory they show up to be
equivalent characteristics of the amount of matter.
Spacetime geometry. 4-vectors, 4-tensors.
In
pre-relativistic physics, space and time act as independent concepts for
describing the motion of bodies. However, STR shows that in
reality, space and time are inextricably intertwined. Lorentz
transformations "mix" the time coordinate with the
spatial coordinates as they move from one frame of reference to
another. A physical quantity, for the measurement of which only ruler is enough for
one observer, the
other observer must measure using a ruler and a watch. The
four-dimensional space-time, which we introduced at the beginning
of this paragraph, thus ceases to be only a formal model, but
acquires a deep geometric-physical
meaning. To clarify this meaning, a metric needs to be introduced in
space-time, ie to define the spacetime "distances" (remoteness)
between events.
Spatio-temporal
interval and metric
An important property of the distance l = Ö[(x_{2}-x_{1})^{2}
+(y_{2}-y_{1})^{2} +(z_{2}-z_{1})^{2}] two points (x_{1},
y_{1}, z_{1}), (x_{2}, y_{2} , z_{2}) in three-dimensional Euclidean
space is its immutability (invariance)
when transitioning
to another system of spatial coordinates (for example, when
shifting or rotating coordinate axes). We have shown above that
the quantity s defined in (1.66) retains its
value in any inertial system, with arbitrary Lorentz
transformations of spacetime coordinates. Invariant quantity s defined relation
s_{1,2 }^{2} = -c^{2} (t_{2} -t_{1}) ^{2} + (x_{2} -x_{1}) ^{2} + (y_{2} -y_{1}) ^{2} + (z_{2} -z_{1}) ^{2} | (1.82) |
and called the space-time interval between events (t_{1},x_{1},y_{1},z_{1}) and (t_{2},x_{2},y_{2},z_{2}), thus
plays the role of space-time distance -
remoteness - of two
events *).
*) The space-time interval s
and its differential element ds play a key role in the
theory of relativity. It expresses how space-time events are
"far" fron each other - in space and time. According to
our conventional notions, two events can be "far apart"
for two reasons :
1. Either they took place in different distant
places in space;
2. Or they happened at different times, there
was a long "time interval" between them.
^{ }The theory of relativity
"mixes" space and time and combines them into a single space-time
continuum. The spatio-temporal "distance"
between the two events "1" and "2" is then
expressed by a space-teme interval s_{1,2} according to Equation (1.82). Something like a
Pythagorean theorem, generalized to 4-dimensional (pseudo)
Euclidean spacetime. This value of the space-time interval does
not depend on the reference or coordinate system with which it is
determined (it follows from the constant speed of light c; and
from this then follows the Lorentz transformations (1.69) STR
derived above) - it is completely objective.
In STR we ussualy suffice with a macroscopic expression of the
interval s, resp. s^{2}. In the following section 2 (as well as in all other
chapters of this book), we see that in the curved space of
general relativity is necessary to use the differential
element of interval ds (resp. its square ds^{2}), which has a special
function expressions characterizing curvature of
spacetime - that in different places there are different
spatial scales and different rates of flow of the (coordinate)
time.
If we know the space-time interval, ie the dependence of the
element ds^{2}
on the coordinates, we know "everything" about
space-time and we can use it to study how bodies (particles) will
move in it and light (photons) will propagate in it, as well as
various physical fields. In other words, we know the metric
tensor g_{ik} and the equations of geodetic lines -
trajectories of free particles in the gravitational field (§2.4
"Physical laws in curved spacetime").
^{ }In this way we have the so-called Minkowski metric introduced in space-time, which we can
write in differential form^{
}
ds ^{2} = -c^{2} dt ^{2} + dx ^{2} + dy ^{2} + dz ^{2} ;
if we introduce a new notation x° º ct, x^{1} º x, x^{2} º y, x^{3} º z, the Minkowski metric will have the form *)
ds ^{2} = - (dx°) ^{2} + (dx^{1} ) ^{2} + (dx^{2} ) ^{2} + (dx^{3} ) ^{2} . | (1.83) |
It differs from the
normal Euclidean metric by a negative sign at the time
coordinate. Such a metric is called pseudoeuclide. While in Euclidean geometry the
distance between two points is zero only when both points merge,
the interval between two events in space-time can be zero even if
the two events are very far apart (eg one
such event can be the transmission of a radio signal here on
Earth and the second event caused by it, the space rocket
maneuver, perhaps somewhere near Jupiter).
*) In the special theory of
relativity (especially in the older
literature) the imaginary
time coordinate x^{4} = i
c.t is
often used, which was introduced by Minkowski to make the
geometry of spacetime formally similar to the geometry of
Euclidean space: ds^{2} = (dx^{4})^{2} + (dx^{1})^{2} + (dx^{2})^{2} + (dx^{3})^{2}. This
formalism has at the geometric intetepretation of STR some advantages, e.g. Lorentz transformation can be represented as a rotation
of the coordinate system.
However, the use of an imaginary time coordinate also has
disadvantages. It obscures
some important structural properties resulting from the pseudo-euclidean character of spacetime,
and operations with complex numbers are unnecessarily used to
calculate some physical quantities that are real. Mainly, however, the use of an imaginary time coordinate loses any significance in the general
theory of relativity, where the geometry of curved spacetime cannot be "make a similar" to Euclidean geometry.
And since STR serves here as a basis for building a general
theory of relativity and studying the general properties of
spacetime, we will fundamentally
use the real time coordinate x° = c.t .
Since the STR engaged movements at speeds close to the speed of light, it is useful in space-time diagrams on the time axis, instead of simply time t plot the time coordinate x° = c.t, so that the scale on the time axis is commensurable with the scales on the spatial axes. Such a space-time diagram, on which the x-axis is marked and the time axis perpendicular to it (the y and z coordinates are omitted for simplicity), corresponding to the initial reference system S, is shown in Fig.1.5c. On these coordinate axes it is possible to read the space-time coordinates of any world point (event) in the reference system S. In order to be able to read the space-time coordinates of these events in the reference system S', moving with respect to S in the direction of the axis x by the speed V, the coordinate axes x' and x'° = ct' corresponding to the system S' must be plotted in this diagram. The x' axis, which is given by the condition t' = 0, is according to (1.69) the line ct = (V/c).x; the axis t', given by the condition x' = 0, is the line x = (V/c).ct. Thus, as can be seen from Fig.1.5c, the transition to another inertial system by means of Lorentz transformations geometrically means the transition to an oblique system of space-time coordinates, the axes of which are inclined with respect to the original axes by an angle a given by tg a = V/c. This angle a increases with the velocity of the system S' relative to S, and at V ® c approaches 45°, when the axes x' and ct' coincide. From such a geometric expression of the Lorentz transformation, the kinematic effects of STR, such as the contraction of lengths or the dilation of time, follow very clearly; the well-known paradox of the clock is also elegantly solved here [232], [242] - it is analyzed in more detail above in the passage " Paradoxes STR " .
Causal relationships in spacetime
Spatial and temporal relationships between events and bodies are
expressed by geometric relationships between the relevant formations
in four-dimensional spacetime. The simplest geometric objects in
space-time are the already mentioned worldpoints representing
individual elementary events.
^{ }The basis of our
cognition of objective reality are causal relationship between phenomena and events.
Let us therefore look, what limitations on the causal
relationships between events give the
regularities of STR.
Let us observe two events As(t_{A},x_{A},y_{A},z_{A}) and Bs(t_{B},
x_{B},y_{B},z_{B}) in terms of the reference
system S (Fig.1.6a). We denote the time interval
between them t_{AB} = t_{B} - t_{A} and their spatial distance l_{AB}: l_{AB}^{2} = (x_{B}-x_{A})^{2} +(y_{B}-y_{A})^{2} + (z_{B}-z_{A})^{2};
the space-time interval s_{AB} between them will be s_{AB}^{2} = -c^{2}t^{2}_{AB} + l^{2}_{AB}. Event B can have some causal connection
with event A only if these events can be connected by
a signal propagating more slowly than light, ie provided that l_{AB} < c.t_{AB} , or
s_{AB }^{2} < 0 .
The interval satisfying
this inequality is called temporal (of the time type, "time-like"). Whether two events
connected by a time-type interval are actually related depends on
the specific circumstances - but in principle they always can.
^{ }If there is an interval between two events of a
temporal nature, it is always possible to find such a reference
system S' in which both events take place in the
same place of the space (l'_{AB} = 0). The time interval between
both events in this system is then t'_{AB} = Ö(-s^{2}_{AB}/c^{2}) > 0. If the interval between
two events A and B is of a temporal nature and from
the point of view of the reference system S
event B
occurred later than A , ie t _{B} > t _{A} , this time relation also
applies in every other inertial system (there is no frame of
reference, in which event B precedes event
A ) - event B is therefore absolutely future with respect to A . If two events A
and B take place with the same body, the interval between
them is always of the time type, because the path l_{AB}, which the body runs between the two
events, is always less than c.t_{AB} (the velocity of the body cannot
be greater than c ), so s^{2}_{AB} = l^{2}_{AB} - c^{2}t^{2}_{AB} < 0.
^{ }If, on the other hand, two events
A and C are separated by an interval
satisfying the inequality^{ }
s ^{2}_{AC} > 0 - spatial type interval ,
is l_{AC} > c.t_{AC} , so between these events can be no causal link (event A was not able to "let itself know" to the event C, because the event C occurred before they could overcome any signal the distance l_{AC}). For every two events A and C separated by a spatial type interval, it is always possible to find a reference system S' in which t'_{AC} = 0, i.e. in which both events take place simultaneously; the spatial distance of this two events is equal to l_{AC} = s_{AC}. In addition, if in system S, event C occurred later than A (t_{C} > t_{A}), there is a reference frame S', from whose point of view of both the time sequence of events opposite: t'_{A} > t''_{C} . At the same time, there is no frame of reference in which such events A and C are same-place coexistent - the events separated by the spatial type interval are therefore absolutely distant from each other.
Fig.1.6. Causal structure and motion of particles in Minkowski
spacetime of special theory
of relativity.
a )
Spatio-temporal diagram of three events A, B, C. Event B
can be causally related to A
(s^{2}_{AB} <0), while event C cannot depend on event A in
any way (s^{2}_{AC} > 0).
b ) Mass
bodies (particles) move in space-time along time-type world lines lying inside light cones, light
propagates along isotropic light lines lying on the mantle of a
light cone, hypothetical tachyons describe space-type world lines.
c) For each
world point, the light cone divides spacetime into causally
related areas of the absolute future and past and into absolutely
distant areas without causation.
One-dimensional curves - world-lines - in 4-dimensional spacetime represent particle motions. Since the speed of each material body is limited by the speed of light, on the space-time diagram of the worldline of each particle it will form an angle of less than 45° with the time axis x°; the set of all worldlines of test particles passing through a given point O thus fills in spacetime the "cone" (4-cone)
x ^{2} + y ^{2} + z ^{2} - c ^{2} .t ^{2} < 0
with a vertex at this
point O according to Fig.1.6b (where
event O is
taken as the origin of the coordinate system). Such worldlines
are called of the time type, because the interval between
their two arbitrary worldpoints (t, x, y, z) and (t+dt, x+dx, y+dy, z+dz) satisfies the relation ds^{2}
<0 - is of time character.
^{ }The photon moves along the world
line dx^{2} = (dx°) ^{2}, ie along a line inclined 45°
to the time axis. The set of worldlines of all photons passing through point O (ie emitted from point O or coming to point O)
forms a "surface" (hypersurface) in space-time^{ }
x ^{2} + y ^{2} + z ^{2} - c ^{2} .t ^{2} = 0 ,
ie the mantle of said
cone - the so-called space-time light cone diverging from point O on all sides at an angle
of 45° to the time
axis x°. This mantle of the light cone in space-time expresses
the propagation of a spherical light wave emanating from the
origin O (x = y = z = 0) at time t = 0. The world lines
lying on the mantle of the light cone are called light, isotropic or zero word lines; the space-time interval between any of
their worldpoints is equal to zero: ds = 0.
^{ }The light cone directed from a given event
O to the future contains all events
that can be an event O affected; light beam converging at a
point O from the past includes all
events that could have an event O influence. The set of all double light
cones emanating from each point (event) of spacetime creates a
branching causal structure in it The respective light cone
for each event (world point) divides spacetime into three areas
(Fig.1.6c): the area of absolute future
and absolute past inside the light cone, and the area
outside it containing "absolutely distant"
events without causal connection.
^{ }In space-time, we can also
imagine the spatial-type world lines, which lies outside
the light cone and the interval between this world points ds^{2}
> 0 is spatial in nature. Spatial-type
world lines represent motion at superlight speed and therefore
cannot correspond to any real body. They could express the
movement of hypothetical tachyons (see below). The
movement of world lines of the spatial type is accompanied
by "pathological" kinematic and causal behavior: on the
space-time diagram it is easy to find a reference system in which
such a particle will be in two different places, and systems in
which the tachyon reaches its target before its source radiated -
violates causality (although there is an intinterpretations in which the
violation of causality does not occur, but there are also certain
problems [102]). In
the following, we will therefore not ascribe physical
significance to spatial-type worldlines. But we will include here
a brief passage about tachyons :
Tachyons
- particles faster than
light ?
It follows from the special theory of relativity, that no
material body or particle can move faster than light, while only
particles with zero rest mass move at the speed of light.
However, some physicists did not want to accept this limitation
and asymmetry in the region of velocities and expressed the
speculative hypothesis that there could be exotic particles
called tachyons (Greek: tachyos = fast ), which would move faster than light [80],
[102] *).
*) Proponents of the tachyon hypothesis divide particles into
three types: Particles with a (real) non-zero rest mass moving at
sublight speed are called bradyons or tardyons. Particles
with zero rest mass moving at the speed of light are called luxons.
And particles that would move at super-light speeds are generally
called tachyons.
^{ }From the basic relations (1.75)
and (1.81) of relativistic dynamics between (inertial) mass,
velocity, momentum and energy, some unusual "exotic"
properties of tachyons follow. The relation m
= m_{o}/Ö(1 - v^{2}/c^{2}) at v > c gives the imaginary
mass of the tachyon; the same is true for its energy E.
If we accelerate the tachyon, its energy decreases; a
zero energy tachyon would move infinitely fast. From the point of
view of quantum physics, the problem would be that in the
formation of virtual pairs of tachyons, they would move farther
apart from each other very quickly than Compton distance and
could not annihilate - the vacuum would become completely
unstable. If the tachyon were electrically charged, it would
perhaps emit Cherenkov's electromagnetic radiation
as it moved through the vacuum at superlight speed *) - this
would reduce its energy and thus increase its speed, the
electrically charged tachyons would spontaneously radiate all
their energy. Even with electrically uncharged tachyon, according
to the general theory of relativity, it can be expected that when
moving through a vacuum at a speed greater than c the
tachyon should emit gravitational Cherenkov radiation
(creating a cone running behind it), which would carry away the
energy of the tachyon, which would thus accelerate to an ever
higher speed.
*) Cherenkov radiation is
electromagnetic radiation generated when an electrically charged
particle moves in an optical medium at a speed exceeding the
speed of light in that medium (which is less than c ).
This radiation is a kind of "shock wave" similar to an
acoustic bang in the atmosphere of an aircraft moving at
supersonic speed. The physical mechanism of Cherenkov radiation
is described in the passage "Cherenkov radiation" §1.6 "Ionizing radiation" of the book
"Nuclear physics and physics of ionizing radiation". "Classical" Cherenkov radiation is
caused by interference of depolarizing electromagnetic waves of
the material environment from individual parts of the particle
path. However, in the case of tachyon in vacuum the material
environment is missing, perhaps there could be electrical
polarization of the vacuum, whose" virtuality "
polarization would become real..?..
Because like electrodynamics accelerated movement of the electric
charges generated electromagnetic waves, according to the general
theory of relativity resulting accelerated movement of the mass
gravitational wave propagating velocity also c , can be
expected gravitational analogy of Cherenkov
radiation (this is author of this book skeptical - by what
mechanism would partial waves be aroused ..?..).
^{ }These "wild" dynamic properties
of tachyons, as well as the kinematic and causal pathologies
mentioned above, are difficult to accept from a physical point of
view. Therefore, the real existence of tachyons in physics is
generally rejected. No phenomena indicative of
the participation of tachyons have been observed, these particles
have no role in the logical structure of theoretical physics,
they are not necessary to explain any phenomenon observed so far.
According to the principle of Occam's razor (discussed
in §1.1), it is therefore assumed that they do not exist.
^{ }Tachyons sometimes appear as some
solutions in the formalism of unitary field theories, cf. §B.6
"Unification of fundamental interactions. Supergravity.
Superstrings.". The classification
of tachyons among other "exotic" and hypothetical
particles in the systematics of elementary particles is mentioned
in §1.5 "Elementary particles", passage "Hypothetical and model particles" of the book "Nuclear
Physics and Physics of Ionizing Radiation".
___________________________________________
Due to the invariance of the interval, the classification of spacetime intervals between events and the particle line of particles into temporal, isotropic (zero) and spatial, as well as the division of spacetime regions according to a causal connection into absolutely future or past and absolutely distant, has absolute meaning, independent of the reference system. Although the specific spatial and temporal relations between events generally depend on the frame of reference from which they are observed, for causally related events, the terms "sooner" and "later" have absolute meaning. Only in this way can the concepts of cause and effect make sense. The theory of relativity thus physically concretizes the concept of causality based on the properties of the propagation of interactions. The connections between causality and the structure of spacetime will be elaborated in more detail in §3.2 and 3.3.
Fig.1.7. Expression of evolution and motion of bodies in
four-dimensional spacetime.
a ) Solid
body T in
three-dimensional space and its projection into the XY plane.
b ) Hypersurface x° = const. = ct in four-dimensional
space-time represents the whole infinite three-dimensional space
at time t_{o} .
c ) Body T describes ("cuts out") a
four-dimensional "world tube" as it moves in
space-time.
d ) World
tube of a pulsating body.
Other geometric shapes
in space-time are two-dimensional surfaces and three-dimensional hyperareas ("supersurfaces"). Hyperplane x° =
const., ie t = const. = t_{o}
in space-time is actually the whole infinite three-dimensional
space in time t = t_{o}. If we have some
(three-dimensional) solid body T (Fig.1.7a) at time t_{o},
it will be expressed in space-time as the corresponding bounded
shape in the hyper-plane x° = ct_{o} = const. (Fig.1.7b), whose
(two-dimensional) boundary represents the surface of the body T at
time t = t_{o}. Physical system of finite dimensions (eg
interior of a body T ) in its movement and
development, it describes ("cuts out") in space-time a
kind of four - dimensional "tube" called space-time or world
tube, which expresses the set of all points of the system
(body) at all times t (Fig.1.7c). The
three-dimensional "mantle" of this tube represents the
surface of the body at all times - the evolution of the shape of
the body. E.g. the surface of a spherical body of constant radius
R with the center at the origin of the
coordinates (ie spherical surface x^{2} + y^{2} + z^{2} = R^{2} = const.) at all times t
will form a cylindrical hyperplot with x-axis in space-time.
^{ }An important special case of the
4-dimensional space-time (world) tube is^{
}light cone,
analyzed above in the section "Causal relationships in
space-time", Fig.1.6. Its three-dimensional
mantle given by the equation x^{2} + y^{2} + z^{2} - c^{2} t^{2} = 0 (light
"hypercone") represents the surface of an ordinary
sphere (light signal wavefront) with a center at the beginning,
the radius of which first decreases with the speed of light from
infinity to to zero, and then increases from time
to speed c to
infinity. Usually, however, only the half of the light cone that
points to the future is taken.
Like worldlines, space-time hyperplanes are classified into
spatial, isotropic (light), and temporal, depending on whether
the square of the interval between their worldpoints is always
positive, can be zero or negative. E.g. hyper-plane t =
const. is a spatial type, the mantle of the light cone is an
isotropic hyperplate.
Four-dimensional vectors and tensors
Spatio-temporal coordinates and components of
quantities in spacetime will be denoted by Latin indices i, j, k,
.., m, n, ..., which take the values 0,1,2,3; eg x^{i} º (x°, x^{1}, x^{2}, x^{3}). We will provide purely spatial
coordinates and components with Greek indices a, b, ...., m, n, ..., running values 1,2,3; eg x^{a} º (x^{1},
x^{2}, x^{3}). When writing algebraic
operations with these indexed variables, the advantage is very conveniently
using so-caled Einstein's summation rule,
according to which addition is performed over each index
occurring twice in the product, whereas the summation symbol S being omitted. For example _{i=0}S^{3}A^{i}A_{i} = A°A_{o}+A^{1}A_{1}+A^{2}A_{2}+A^{3}A_{3}
s A^{i}A_{i}; simplification of writing formulas is evident.
The expression for the space-time interval (1.83) STR is a special case of the general quadratic form
ds ^{2} = g _{ik} dx ^{i} dx ^{k} = h _{ik} dx ^{i} dx ^{k} , | (1.84) |
whose coefficients, the so-called metric tensor g_{ik} (see §2.1) *), have a special simple shape
g _{ik} = h _{ik} º | / | -1 | 0 | 0 | 0 | \ | ; | ||||
| | 0 | 1 | 0 | 0 | | | ||||||
| | 0 | 0 | 1 | 0 | | | ||||||
\ | 0 | 0 | 0 | 1 | / |
h_{ik} is sometimes called the Minkowski metric tensor.
*) Here in STR, the introduction of the metric tensor is only
formal, when using common Cartesian coordinates it has trivial
values of components. In Chapter 2 (and in all others) we will
see that the metric tensor is of key importance in the general
theory of relativity - it describes gravity as the geometry
of curved spacetime.
Transition from inertial system S with coordinates x^{i} s (x°,x^{1},x^{2},x^{3}) to system S 'with coordinates x'^{i} s (x'°,x'^{1},x'^{2},x'^{3}) it must be a linear transformation of spacetime coordinates
x'^{i} = _{k=0}S^{3}a^{i}_{k} x^{k} + b^{i} = a^{i}_{k} x^{k} + b^{i} , i=0,1,2,3 | (1.86) |
( a^{i }_{k} and b^{i} are constants independent of x ), because according to the principle of relativity a particle moving uniformly rectilinearly in the inertial system S must also move uniformly rectilinearly from the point of view of every other inertial system S'. In order to satisfy the principle of constant speed of light, this transformation must further satisfy the condition
s^{2} = h_{ik} x^{i} x^{k} = h_{ik} x'^{i} x'^{k} = s'^{2} | (1.87) |
of interval invariance. The transformations x^{i} ® x^{'i} (1.86) satisfying the condition (1.87) are a four-dimensional expression of the general Lorentz transformations between the inertial systems S and S' . If we measure coordinates and time in such a way that at t = t '= 0 the beginnings of Cartesian coordinates in both systems S and S' coincide, they are b ^{i} = 0 - these are the so-called homogeneous Lorentz transformations
x ^{' i} = a ^{i }_{k} x ^{k} . | (1.86 ') |
In Fig.1.5c we have
shown that the Lorentz transformation geometrically means a
transition between oblique space-time coordinates.
^{ }The transformation relation
(1.86) contains a total of 4 x 4 = 16 seemingly independent coefficients a^{i }_{k}. Substituting from the
transformation relation (1.86') into (1.87) we get the condition h _{ik} = h _{lm} a ^{l }_{i} a ^{m }_{k} , which binds these coefficients
by 10 equations (with respect to the symmetry in the indices i,
k). Therefore, only 6 independent transformation coefficients
remain in (1.86') - they correspond to the three parameters
indicating the direction of the x', y', z' axes and to the three
components of the velocity vector of the system S'^{ }against S . The set of all homogeneous
Lorentz transformations (1.86') forms a group - a continuous
6-parameter Lorentz group ( ¥^{6} ).
^{ }Also, the set of all inhomogeneous Lorentz
transformations (1.86), which arise from homogeneous
transformations by adding four transformations of the shift of
the beginning of space-time coordinates x^{'i} ® x^{' i} + b^{i} , forms a 6 + 4 = 10 -parameter
group - the so-called Poincaré group.
^{ }In the case of a special Lorentz
transformation, the relation (1.86 ') goes to (1.69), so the
coefficients a^{i }_{k} have values
(1.86 '') |
The main task of the
special theory of relativity is the formulation of physical laws
independently of the inertial frame of reference. In
four-dimensional space-time, these physical laws translate into
geometric relationships between objects in space-time that are
independent of the choice of space-time coordinates. Like in the
three-dimensional space of classical physics vector notation of physical
laws guaranteeing their validity independent of the used spatial
coordinates (permanence eg. in shifts or rotation
appreciate the
coordinate axes), fulfilling the principle of relativity in the
STR can be best expressed by the fact that the physical laws are
formulated as vector and tensor
equations in
four-dimensional spacetime. Such a vector or tensor equation
valid in one coordinate system automatically applies in every
other coordinate system. In addition, the laws of mechanics and
electrodynamics take on a particularly simple and illustrative
character when expressed by the relationships between vectors and
tensors in four-dimensional spacetime - see below "Four-imensional mechanics" and "Four-imensional electrodynamics".
^{ }Coordinates (ct, x, y, z) = (x°,
x^{1} , x^{2} , x^{3}) º x ^{i}^{ }the given events can be
considered as components of the four-dimensional "position
vector" of the respective worldpoint in space-time. The
"length" square of this position 4-vector can then be
defined as the interval between the origin (0,0,0,0) and the
given point (x°,x^{1},x^{2},x^{3}): (x^{i})^{2} = -(x°)^{2} +(x^{1})^{2}+(x^{2})^{2} +(x^{3})^{2} = h_{ik} x^{i} x^{k }; it is an invariant quantity
with respect to Lorentz transformations. In the context of the
general definition of vectors in n-dimensional space, a four-dimensional
vector (4-vector) A^{i} means a set of four quantities
A°, A^{1}, A^{2}, A^{3}, which are transformed in the
same way as the coordinates x^{i} during the transformations (1.69
') of space-time coordinates :
A' ^{i} = a ^{i }_{k} A ^{k} = ( ¶ x' ^{i} / ¶ x ^{k} ) . A _{k} . | (1.88) |
In addition to the mentioned components of 4-vectors A^{i} with indices at the top, called contravariant, the so-called covariant components A_{i} with indices at the bottom are also introduced using the relation
A_{i} s h_{ik} A^{k} , eg. A_{o} = -A° , A_{1} = A^{1} , A_{2} = A^{2 }, A_{3} = A^{3} . | (1.89) |
It can be easily shown that the transformation properties of the covariant components are
A' _{i} = ( ¶ x ^{k} / ¶ x' ^{i} ) . A _{k} , | (1.88 ') |
i.e., the covariant and contravariate components transform each other
"contra-gradient".
^{ }The scalar
product of
two 4-vectors A and B means the algebraic expression A^{i}B_{i}
= A°Bo + A^{1}B_{1} + A^{2}B_{2} + A^{3}B_{3} = h_{ik}A^{i}B^{k} = -A°B°+A^{1}B^{1}+A^{2}B^{2}+A^{3}B^{3} = A_{i}B^{i}; it is a scalar invariant with
respect to coordinate transformations. The square of the size of
a given 4-vector A is defined as its scalar product
with himself: (A)^{2} s A^{i}A_{i}
= -(A°)^{2}+(A^{1})^{2} +(A^{2})^{2}+ (A^{3})^{2} . According to the sign of the
square 4-vector space-time four-vector are divided into three
groups: A ^{i} A _{i} <0 - vector
of the time type; A ^{i} A _{i} = 0 - zero or isotropic
vector; A ^{i}
A _{i} > 0 - spatial
type vector.
The three spatial components A^{1}, A^{2}, A^{3} of the
4-vector A^{i} form a three-dimensional vector A (due to transformations of purely spatial
coordinates), so the set of 4-vector components can be
symbolically written as A ^{i} º (A°, A). Such a distribution of the
4-vector into space and time can be done in any inertial system,
but of course it changes with Lorentz transformations. The square
of the 4-vector A^{i} then is A ^{i} A _{i} = - (A°)^{2
}+ A^{2}.
For vector A^{i} of the time type, a system S' can always
be found in which the spatial vector A' = 0 (it is a system S' whose time
axis has the direction of the 4-vector A^{i}); similarly, for each vector B^{i}
of spatial type, a system S' can be
found in which its time component B'° = 0.
In space-time, more complex quantities - tensors - are also introduced by means of their transformation properties. The contravariant 4-tensor of the r-th order means the sum of 4^{r}^{ }of the quantities T ^{i}^{1 }^{, i}^{2 }^{, ..., i}^{ r} , which are transformed during the transformation of the coordinate system x ^{i} ® x' ^{i} = a ^{i }_{k} x ^{k} as a product of r -coordinates x ^{i} :
T'^{i}^{1}^{,i}^{2}^{,...,i}^{r} = a^{i}^{1 }_{k}_{1}. a^{i}^{2 }_{k}_{2} ... a^{i}^{r }_{k}_{r}_{ }. T^{k}^{1}^{,k}^{2}^{,...,k}^{r} .
Analogously covariant
and mixed tensors - see general definition in §3.1. A scalar is
a 0th order tensor, a vector a 1st order tensor.
^{ }The connection between covariant
and contravariant components of tensors, ie "raising"
and "lowering" indices, takes place via the metric
tensor, in STR via the Minkowski tensor h_{ik}. E.g. T_{ik} =h_{im}T^{m}_{k} = h_{il}.h_{km}.T^{lm}. When used Minkowski metric is a
simple rule: when lifting and lowering spatial indexes (1,2,3) the values of components
do not change, when raising and lowering time index ("o")
changes sign this component.
^{ }Arithmetic operations between
tensors (components of tensors) are governed by simple and
natural rules of tensor algebra
[214], [163], [33]. With tensor product
are created tensors
of higher orders, e.g. 2nd order A^{ij }tensor product to B^{k} of the 1st order (i.e.
four-vector) formed 3.order tensor T ^{ijk} = A ^{ij} .B ^{k} ; analogously for mixed tensors.
Conversely, a "narrowing" operation, consisting
of summation over a pair of indices in a given tensor, creates
lower order tensors. E.g. from the tensor of the fourth order A^{iklm} by narrowing the tensor of the second
order A^{ik} = A ^{ikl }_{l
}is formed; by narrowing the tensor of the 2nd order
A^{ik} we get the scalar A = A^{i}_{i} = A°_{o}+A^{1}_{1}+A^{2}_{2}+A^{3}_{3 }, which
is called the trace of the tensor A^{ik}.
^{ }Among the 2nd order tensors, the Minkowski
tensors h_{ik} and h^{ik} occupy a special position, as well as the
so-called Kronecker delta-symbol d^{i }_{k}: d ^{i }_{k} = 1 for i = k, d ^{i }_{k} = 0 for i ¹ k -
its trace d^{i }_{i} = 4; the components of these tensors are
the same in all STR coordinate systems. Such tensors are called isotropic. Applies to h^{im} . h_{mk} = d ^{i }_{k} and for each vector A^{i}
is d ^{k
}_{i} A^{i}
= A^{k} ; tensor d ^{k }_{i} thus has the character of a unit
4-tensor of the 2nd order. In the tensor calculus, a unit
isotropic tensor of the 4th order - Levi-Civites tensor e ^{iklm} antisymmetric in all indices is also often used, whose
component e ^{0123}= +1 and the other non-zero
components (ie those for which all four indices are different)
are equal to +1 or -1 depending on whether the given sequence of
indices i, k, l, m is from the sequence 0,1,2, 3 formed by an
even or odd number of permutations.
If we have scalar,
vector or tensor quantities defined not only at one point, but at
each point of a given area of space (here space-time), we speak
of scalar, vector and tensor fields. The rules and operations of
vector analysis, so useful in physics of field and continuum, are natural
to transfer and generalize to four-dimensional spacetime.
^{ }The 4-gradient of
a scalar field j = j(x^{i}) is defined as a four-vector
whose covariant components are^{
}
(1.90) |
The four divergence of the vector field A^{i} = A^{i} (x^{k}) means a scalar field
A ^{i }_{, i} º ¶ A ^{i} / ¶ x ^{i} = ¶ A ° / ¶ t + div A ; | (1.91) |
analogously, the 4-divergence of the tensor field T^{ik} is a four-vector (vector field) T^{i} = T^{ik}_{, k} º ¶ T ^{ik} / ¶ x ^{k} . It is advantageous to denote the differential operator ¶/¶x ^{i} simply by an index with a comma "_{, i} ", which considerably simplifies the notation of such relations. The operator ¶ / ¶x^{i} is a generalization of the Hamiltonian operator Ñ = i . ¶ / ¶x + j . ¶ / ¶y + k . ¶ / ¶z . The space-time generalization of the Laplace differential operator D = ¶^{2}/¶x^{2} + ¶^{2}/¶y^{2} + ¶^{2}/¶z^{2} is the d'Alembert operator
(1.92) |
Thus žj
= j^{ ,i}_{,i} = ¶^{2}j/¶x^{2} + ¶^{2}j/¶y^{2} + ¶^{2}j/¶z^{2} - (1/c^{2}) ¶^{2}j/¶t^{2}.
^{ }Gauss's theorem of vector
analysis in three-dimensional Euclidean space
(1.93) |
according to which the integral of the divergence of a vector A over some volume V is equal to the flow of this vector over a closed surface S = ¶V bounding this volume, is generalized to the shape in four-dimensional space-time
(1.93 ') |
where dW = dx^{0} dx^{1} dx^{2} dx^{3} = c.t.dV is a 4-volume element in
spacetime and dS_{i} are the 4-vector components of
the hyperplanar element S = ¶W
bounding the
4-volume W , through which it integrates on
the left side; dS° = dx^{1} dx^{2} dx^{3} = dV, dS^{1} = dx^{0} dx^{2} dx^{3} = c.dt.dy.dz, similarly dS^{2}
and dS^{3} .
^{ }Relationship between the curve
integral of a vector over a closed curve C
and the area
integral over the area S, bounded by the curve C,
is given in the three-dimensional vector analysis by the Stokes
theorem
(1.94) |
The integral along the closed four-dimensional curve C is converted to the integral over the hyperplot S bounded by this curve C in general so that dx^{i} is replaced by dS^{ik} ¶/¶x^{i}. A direct generalization of the Stokes theorem for the curve integral of a 4-vector A^{i} then reads :
(1.94 ') |
where the components of the antisymmetric surface tensor dS^{ik} = dx^{i} dx'^{k} - dx^{k} dx'^{i} give the projections of the planar element (taken as a parallelogram with the sides dx^{i} and dx'^{i} ) into the coordinate planes. Analogously for higher order tensors.
Four-dimensional mechanics
In classical
mechanics, the motion of a material particle is described by a trajectory in three-dimensional Euclidean space
r = r(t) , i.e. x^{a} = x^{a} (t), a = 1,2,3 .
In four-dimensional space-time, the motion of a particle is represented by its worldline, which can be described by a parametric equation
x ^{i} = x ^{i} ( l ) , | (1.95) |
where l is a suitable parameter. As a parameter l it is possible to use either the
coordinate time t , but better invariant
quantities - the proper time t
or directly the
"length" of the worldline given by the space-time
interval s .
^{ }The vectors of velocity v = dx/dt
and acceleration a = dv/dt = d^{2
}x / dt ^{2} play an important role in
classical mechanics, so it is useful to introduce their
four-dimensional analogies. The quantity dx^{i}/dt created by direct
generalization is not suitable, because it is not a four-vector
(dt is not an invariant). The invariant measure of time is the
proper time t, so as a four-velocity it
is natural to define a 4-vector with components *)
u ^{i} = ^{def} dx ^{i} / d t = c. dx ^{i} / ds . | (1.96) |
Given the relationship (1.72) between dt and dt, the four-velocity components can be expressed using the ordinary velocity v in the form
(1.96 ') |
at low velocities v «c the spatial part of the 4-velocity changes to ordinary velocity v. From the relation dx^{i}dx_{i} = c^{2}dt^{2 }easily follows
u ^{i} . u _{i} = c ^{2} . | (1.97) |
From a geometric point
of view, a four-vector c.u^{i} is a unit tangential vector to the world line of a
given particle.
*) Often the 4-speed is
defined as u ^{i} = dx^{i}/ds = c^{-1} dx^{i}/dt; the four-speed defined in this way is a
dimensionless quantity.
^{ }4-acceleration of
particle is defined
as^{ }
a ^{i} = ^{def} du ^{i} / d t = d^{2} x ^{i} / d t^{2} = c^{2} d^{2} x ^{i} / ds^{2} . | (1.98) |
From the derivative of relation (1.97) according to t then follows
a ^{i} . u _{i} = 0 , | (1.99) |
i.e., the four-vectors of velocity and acceleration vectors in space-time are perpendicular to each other. Movement of free particles, that runs uniformly and linearly (v = const., a = 0), is expessed in four-dimensional form by the equation
a ^{i} º d^{2} x ^{i} / d t^{2} = 0 | (1,100) |
describing a straight
line.
^{ }By four-dimensional
generalization of momentum p = m_{o} .v of classical mechanics there is a
4-vector^{ }
p ^{i} = ^{def} m _{o} . u ^{i} | (1,101) |
called four-momentum. Substituting the components u^{i} from (1.96 ') we get
By comparing relationships (1.75) and (1.80) for the momentum and energy in the STR is seen that the spatial portion 4-momentum is equal to the relativistic momentum p = m. v = m_{o} . v / Ö(1-v^{2}/c^{2}) and the time component is p° = E/c. The 4-moment components can therefore be written as
p ^{i} = (E / c , p ) . | (1.101 ') |
From the space-time point of view, therefore, the energy E and the momentum p of the particle are components of a single four-vector - 4-momentum, which can therefore be described as a kind of "energy-momentum vector" unambiguously characterizing the state of motion of the particle. According to (1.101), the square of 4-momentum p ^{i} p _{i} = (m_{o} u ^{i} ). (m_{o} u _{i} ) = m_{o} .c^{2} is equal to p ^{i} p _{i} = E^{2} / c^{2} - p.c^{2}, which leads to the relation (1.81).
4-vector of force or 4-force is defined as the time change of the 4-momentum of a particle
f ^{i} = ^{def} dp ^{i} / d t = m_{o} du ^{i} / d t . | (1,102) |
This 4-force components are related to the ordinary three-dimensional force vector F = dp / dt by relation
(1,102 ') |
Between the 4-force and
the 4-momentum, the relation f ^{i} .u _{i} = 0 holds, ie the 4-force is
"perpendicular" to the 4-velocity in space-time.
^{ }Newton equation of
d p / dt = F has a four-dimensional generalization shape
dp ^{i} / d t = f ^{i} ; | (1,103) |
the spatial part of this
equation describes the change in momentum, the time component the
change in energy of a particle under the influence of the applied
force.
^{ }In theoretical
physics, the laws of motion are often derived using the variational
principle of least action [165]. A free (relativistic) particle of rest
mass m_{o}, moving in spacetime from point A or
point B , is described by the integral of the action
where s is the space-time interval and t is the particle's own time. This action S is proportional to the length of the particle world-line, ie the relativistic interval s . The variational principle of the smallest action dS = 0 then leads to Lagrange's equations, from which the equations of motion of relativistic mechanics (1.100 ) follow.
Energy-momentum tensor
The quantities of energy and momentum are used
either as characteristics of individual discrete particles and
bodies, or as aggregate quantities characterizing the system as a
whole. However, if the particles in the investigated system are
distributed densely enough that we can consider them as a
continuum, or even a field (in §1.5 we have shown that the
field is a certain "spread" form of matter), it is
necessary to investigate the density with which basic physical
characteristics such as mass, energy, momentum, angular momentum,
electric charge, etc. are distributed in space. In addition, it
is useful to describe how these quantities flow
from one place to another in the system.
^{ }If we denote the energy density e= dE / dt, the local law of energy
conservation is expressed by the continuity
equation
¶ e / ¶ t + div ( v . e ) = 0 . | (1,104) |
Due to the universal relationship (1.81) between energy, mass and momentum , the density of momentum distribution P = dp / dV is given by the energy current density v.e : P = v . e / c (= v . r for incoherent dust). The local law of conservation of momentum can be written in the form
div (v. P ^{a} ) + ¶ P ^{a} / ¶ t = 0, ( a = 1,2,3) | (1,105) |
(preserves each component P^{a} of
momentum).
^{ }We know that energy and momentum are
components of the 4-vector of energy-momentum (4-momentum) in
space-time. Likewise, equations (1.104) and (1.105) of
conservation of energy and momentum can be combined into one
tensor equation
¶ T ^{ik} / ¶ x ^{k} º T ^{ik }_{, k} = 0 , | (1,106) |
where
T^{ik} = e . | / | 1 | v_{x}/c | v_{y}/c | v_{z}/c | \ | (1.107') |
| | v_{x}/c | v_{x}v_{x}/c^{2} | v_{y}v_{x}/c^{2} | v_{z}v_{x}/c^{2} | | | ||
| | v_{y}/c | v_{x}v_{y}/c^{2} | v_{y}v_{y}/c^{2} | v_{z}v_{y}/c^{2} | | | ||
\ | v_{z}/c | v_{x}v_{z}/c^{2} | v_{y}v_{z}/c^{2} | v_{z}v_{z}/c^{2} | / |
is the energy-momentum tensor .
^{ }The energy-momentum tensor, which
completely describes the distribution
and motion of energy and momentum in a given physical system, generally has
the structure :
T ^{ik} = | / | energy density | energy current density, ie (momentum density) / c |
\ | (1,107) | ||
| | | | ||||||
| | energy current density, ie (momentum density) / c |
momentum current density,
ie pressures and stresses (stress tensor) |
| | ||||
\ | / |
The physical significance of the individual components of the energy-momentum tensor T^{ik} is thus as follows :
By breaking down the tensor law of conservation (1.106) into the component and separating the spatial and temporal derivatives we get the equations
(1/c) (¶T°°/¶t) + ¶T°^{a}/¶x^{a} = 0 , (1/c) (¶T^{a}°/¶t) + ¶T^{ab}/¶x^{b} = 0 .
By integrating them over some (arbitrary) spatial area V:
(1/c) ¶/¶t _{V}nT°° dV+ _{V}n(¶T°^{a}/¶x^{a}) dV = 0 , (1/c) ¶/¶t _{V}nT^{a}° dV+ _{V}n(¶T^{ab}/¶x^{b}) dV = 0
and after adjusting with a Gaussian theorem (three-dimensional), two equations are created
(1,106 '') |
wherein the integral on the right side they are taken across the closed surface S = ¶V surrounding volume V. According to the first of these equations, the rate of change of the energy contained in the volume V is equal to the flow of energy through the closed area S bounding this volume. Similarly, the second equation says that the momentum contained in volume V per unit time is equal to the flow momentum through the boundary surface S . Equations (1.106'') express the law of conservation of energy and momentum in integral form. In general, the total 4-momentum p^{i} is expressed by an integral using the energy-momentum tensor
p ^{i} = (1 / c) ò T^{ik} dS _{k}
across a hypersurface
covering the entire three-dimensional space. Equation (1.106) is
then equivalent to the assertion that this 4-momentum
is preserved.
^{ }The ordinary (three-dimensional ) angular momentum vector J of classical
mechanics, defined as J = r ´ p (vector product), is replaced in
STR by a more general 4-angular momentum
tensor
J ^{ik} = x ^{i} p ^{k} - x ^{k} p ^{i} .
It antisymmetric tensor whose components are equal to the spatial components of a three-dimensional vector J. For the continuum, J^{ik} = n(x^{i}dp^{k} -x^{k}dp^{i}) = (1/c) n(x^{i}T^{km} - x^{k}T^{im})dS_{m}. For the law of conservation of the angular momentum J^{ik}_{, k} = 0 to apply, it must be (x ^{i} T ^{km} - x ^{k} T ^{im} ) _{, m}= 0; in addition to the law (1,106), it is necessary for the energy-momentum tensor to be symmetric (T^{ik} = T^{ki}).
The simplest type of substance - of the continuum - is a set of non-interacting particles called incoherent dust. The density of 4-momentum of particles in such a system is then r .u ^{i} , so T°^{a} = r .c^{2} u ^{a} ( a = 1,2,3). The energy density is T°° = r .c^{2} and density the momentum current T^{ab} = r .c.dx ^{a} / dt dx ^{b} / dt . Thus the energy-momentum tensor for incoherent dust is
T ^{ik} = r . u ^{i} u ^{k} .
If we use a reference
system in which the considered volume element is at rest when
monitoring the ideal fluid, Pascal's
law will apply
according to which the pressure p is the same in all directions.
In such a reference frame the stress tensor will be equal s ^{ab} = p. d ^{ab} = T ^{ab }; the momentum density is equal to zero
here, so T°^{a} = 0, and the energy density
T°° = e = r .c ^{2}
.
^{ }The energy-momentum
tensor of an ideal
fluid in the rest frame is therefore
T ^{ik} = | / | e | 0 | 0 | 0 | \ | . | ||
| | 0 | p | 0 | 0 | | | ||||
| | 0 | 0 | p | 0 | | | ||||
\ | 0 | 0 | 0 | p | / |
After the transformation into a general frame of reference in which a given volume element moves with a four -velocity u ^{i} , the energy-momentum tensor of an ideal fluid will be given by
T^{ik} = (p + e) u^{i }u^{k} + p.h^{ik} , resp. T^{i}_{k} = (p + e) u^{i }u_{k} + p.d^{i}_{k} . | (1.108) |
It is advantageous to maintain the concept of the energy-momentum tensor even if it is not a real continuum. If the investigated system consists of N "point" parts of masses m_{a} (a = 1,2, ... , N), which move at 4-velocities u ^{i }_{a} , then the energy-momentum tensor can be defined as
where d^{3}(x) is a three-dimensional Dirac delta-function.
Four-dimensional electrodynamics
Maxwell's
equations of electrodynamics, although originally based on
classical non-relativistic ideas, are invariant with respect to
the Lorentz transformation. Electrodynamics is therefore fundamentally relativistic - electromagnetic phenomena are actually the only
case where we encounter relativistic effects in everyday life (but it is not easy to notice!). Electrodynamics therefore does not need
any relativistic reformulation, the theory of relativity does
not lead to any new and unexpected electromagnetic phenomena. However,
the application of the laws of the special theory of relativity
introduces a clearer and more uniform order in electrodynamics
and points to the deep internal
context of phenomena, which are classically understood as
independent empirical facts. This unity of electromagnetic laws
stands out especially clearly in a four-dimensional space-time
formulation. The basic quantity of electrodynamics is the electric charge, for which the law of conservation
applies (1.31). The electric charge is defined as an invariant scalar quantity, so the magnitude of the charge
of a body is the same in all inertial frames of reference :
dq ' = r '. dV = r dV º r . dx^{1} dx^{2} dx^{3} = dq .
Since the volume is transformed according to the relation dV' = Ö(1 - V^{2}/c^{2}) dV during the transition to another inertial system, the transformation law for r is the same as for dx°: r' = r /Ö(1 -V^{2}/c^{2}). Thus, the density of an electric charge is transformed as a time component of a four-vector. Current density vector components j = r . v , which are j^{a} = r .v^{a} = r .dx^{a}/dt (a= 1,2,3), due to the behavior r are transformed as dx^{a}, ie as spatial components of the four-vector. It is therefore natural to unify the quantities r and j into one 4-vector j ^{i} º (c. r, j), the so-called four-current, the components of which are
j° = c.r , j^{1} = j_{x} , j^{2} = j_{y} , j^{3} = j_{z} . | (1.109) |
Since the component j° = c. r can be expressed by x° = ct as j° = r .dx°/dt, the 4-stream components can be defined as follows :
j ^{k} = r . dx ^{k} / dt . | (1.109 ') |
The continuity equation (1.31b) ¶r/¶t + div j = 0, expressing the law of conservation of electric charge, can then be written in four-dimensional form
¶ j^{ k }/ ¶x^{ k} = 0 , or j^{ k }_{,k} = 0 | (1.110) |
(4-divergence of four-currents
is equal to zero).
^{ }Similarly, for the potentials from
equations (1.46a, b) it follows that in terms of transformation
properties the quantity j behaves as time and the
quantities A^{a} = ( A
) as spatial components
of the 4-vector, so the electric scalar potential j and the magnetic vector potential A can be unified into a single 4-vector^{
}
A ^{k} º ( j , A ) , | (1.111) |
which is called the four-potential. Equations (1.46a) and (1.46b) can then be combined into one space-time equation
ž A ^{k} º ¶^{2} A ^{k} / ¶ x ^{m} ¶ x _{m} = - (4 p / c). j ^{k} , | (1.112) |
wherein the Lorentz calibration condition (1.45) in four-dimensional form
¶ A ^{k} / ¶ x ^{k} º A ^{k }_{, k} = 0 | (1.113) |
indicates that the 4-potential is chosen so that its four-divergence is equal to zero. The development of the relations E = -gradj - (1/c)¶A/¶t , B = rot A between the potentials and intensities of the electric and magnetic field shows, that the components of the vectors E and B can be interpreted as components of the antisymmetric 4-tensor F^{ik}
F ^{ik} = ^{def} ¶ A ^{k} / ¶ x ^{i} - ¶ A ^{i} / ¶ x ^{k} , | (1.114) |
which is called the electromagnetic field tensor. This electromagnetic field tensor, expressed by the components of the vectors E and B , has the structure :
F ^{ik} = | / | 0 | E _{x} | E _{y} | E _{z} | \ | . | (1.114 ') |
| | -E _{x} | 0 | 0 | 0 | | | |||
| | -E _{y} | B _{z} | 0 | -B _{x} | | | |||
\ | -E _{z} | -B _{y} | B _{x} | 0 | / |
It is a unifying
"conglomerate" of electric and magnetic field
components, that completely describes the
electromagnetic field in four-dimensional spacetime.
Note:^{ }The
electromagnetic field tensor F^{ik} is sometimes called the Faraday tensor,
although no tensors (and not 4-dimensional
at all!) were not known in Faraday's time. It reflects the unification
of the electric and magnetic fields in the spirit of
Faraday's law of electromagnetic induction.
^{ }Lorentz equations of motion (1.30) of a
charged mass particle in an electromagnetic field
can be interpreted as spatial components of the four-dimensional equation of motion of a charged particle
m_{o} du ^{i} / dt = (q / c) F^{ik} u _{k} ; | (1.115) |
the time component of
this equation describes the changes in the energy of the particle
as a result of the work performed by the electromagnetic
forces.
^{ }The first pair of Maxwell's equations
(1.40) - (1.41) can be written as one equation for the components
of the electromagnetic field tensor :
(1.116a) |
The second pair of Maxwell's equations (1.38) - (1.39) can then be unified into a single four-dimensional equation using F^{ik}
(1,116b) |
describing the excitation of the
electromagnetic field
by four-currents j ^{i} .
^{ }To formulate electrodynamics
using Hamilton's least action principle, the Lagrangian
of the
electromagnetic field is in a 4-dimensional form:^{ }
L = (1/16p c) F ^{ik} F _{ik} + (1 / c^{2} ) A _{i} j ^{i} ; | (1,117) |
from the variational
principle d S = d ò L d W
= 0 (with variation of the 4-potential components at the
given distribution and motion of charges) we can then obtain Maxwell's equations of
the electromagnetic field (116a, b).
^{ }The relations (1.52) - (1.56),
expressing the density and current of energy and momentum in an
electromagnetic field, can be summarized using the
energy-momentum tensor of the electromagnetic field, which is
equal to^{ }
T^{ik}_{elmag.} = -^{ 1}/_{4}_{p} F^{i}_{m} F^{km} + ^{1}/_{16}_{p} F_{lm} F^{lm} . | (1,118) |
After substituting the values F^{ik} from (1.114') it can be seen, that T°°_{elmag. }is equal to the energy density (1.52) and the components T°^{a }_{elmag. }are equal to the components of the Poynting vector (1.55). Spatial components
T^{ ab}_{elmag.} = s_{ab} = ^{1}/_{4}_{p} [^{1}/_{2}(E^{2} + B^{2}) d_{ab} - E_{a}E_{b} - B_{a}B_{b}]
forms a three-dimensional tensor called the Maxwell stress tensor.
Thus, for mechanical and electrodynamic phenomena, the special theory of relativity is perfectly elaborated and experimentally verified. However, it must be admitted that for other than electromagnetic phenomena, the special theory of relativity is not directly verified... However, indirect indications are very convincig..!..
Nonlinear electrodynamics
At all intensities we observe in nature and in the laboratory,
the electric and magnetic fields in vacuum appear to us to be linear
- for the values of intensities E and B
, as well as for potentials, the principle of
superposition applies exactly. The question is, what
about extremely strong electromagnetic fields?
Variants of generalized nonlinear electrodynamics NED
were constructed for this situation, in which the principle
of superposition does not apply (the
resulting electromagnetic field of two charges is not equal to
the sum of the fields of individual charges).
There is self-interaction field - there is, for
example, the scattering of "light on light". It is
expected that these phenomena of nonlinear electrodynamics *)
could manifest themselves in the field of very strong
electromagnet. fields and high-intensity beams of electromagnetic
radiation (not yet achieved in our
experiments).
*) We mean the behavior of fields in
vacuum, not the phenomena of nonlinear optics
caused by nonlinear polarization or magnetization in material
environments.
^{ }Motivation for the generalization of
classical electrodynamics began to appear at the beginning of the
20th century in connection with a not entirely satisfactory model
of a charged particle (a point charged
particle in the center showed infinite values of the Coulomb
field, with infinite energy - singularity ). In 1912, G.Mie suggested that the electromagnetic
field could be composed of the sum of the classical Maxwell field
and a second nonlinear additional term (composed of potentials A ),
which would be significant only in the area of atomic dimensions (this variant did not penetrate into the wider physical
consciousness public).
Some other modifications of
Coulomb's law for intensity E and potential F were also tried :
- Small correction "e" in the law of
inverted squares: E(r) = Q / r ^{2+ }^{e} ;
- To the standard Coulomb potential F(r) = Q / r include
componentr wit Yukawa potential F'(r) = Qc ^{- }^{m }^{.r}
/ R (used in nuclear physics - is mentioned
e.g. in the passage "Strong
nuclear interactions"
§1.1 monograph "Nuclear Physics and ionizing
radiation"). This would be equivalent
to the introduction of a non-zero photon mass m_{f} with a Compton length
m = m_{f} .c / h ......
None of these initial attempts at nonlinear extensions of
electrodynamics led to satisfactory results...
^{ }Basic consistent variant of nonlinear
generalization of classical Maxwell's electrodynamics
NED was designed in 1934 by M.Born and L.Infeld [23] - Born-Infeld
electrodynamics (BI). The motivation was to remove the
singularity in the point charge - in order to determine the final
value of the electron's own electric energy. For this purpose,
they introduced the hypothesis of the maximum possible
value of the electric field intensity E_{max} and the potential F_{max} , analogously to the relativistic mechanics STR there
is a maximum possible (limit) speed c of motion of
material bodies. They realized this by introducing a special parameter
of nonlinearity "b" in the
modified field equations so that the intensity of the electric
field of the point charge Q (instead of the
standard Coulomb's law E = Q / r^{2} ) depended on the distance r according to the law :
E(r) = Q / Ö(r ^{4} + Q ^{2} .b ^{2} ) |
(dotted curve in Fig.1.8 on the right), at which the intensity of the electric field at the
beginning r = 0 is finite and also the total
energy of the electric field is finite (relations
(1.122-123)).
Note:^{ }Sometimes
the parameter "b" is denoted by the Greek "b" and its inverse
value 1/b or 1/b .
^{ }In classical Maxwell's electrodynamics,
the Lagrangian in a 4-dimensional formulation is given by the
above relation (1.117) :
L = (1/16p c) F
^{ik} F _{ik} + (1 / c ^{2} ) A _{i} j ^{i} .
Lagrangian of the electromag. field in BI is modeled in a
generalized form containing the nonlinearity parameter
"b" (which then characterizes the
maximum possible electric field intensity in the equations - see relations below (1.123) ) :
L_{BI} = b^{-2} [(1 - Ö[1 + b^{2}.F_{ik}F^{ik}/2 - b^{4}.(F_{ik}F^{ik}e_{iklm})^{2}/16]] + A_{i} j^{i} , | (1.119) |
where F_{ik} is the tensor of the electromagnetic field (1.114) (F = F .. ... e Iklm. .. his ..associated multiplied unit ...
Levi-Civitùv tensor e ^{Iklm} antisymmetric. .. add, modify , in Lagrangians, we
are omitting c (convention c = 1) ............ ).
For a pure electric field E, i.e. when B = 0, is
L = b^{-2} [1 - Ö(1 - b^{2}.E^{2})].
^{ }In 3-dimensional vector
symbolism of electric intensity E and magnetic
induction B is the standard Maxwell's lagrangian
(1.42): L = 1/8p (E^{2} - B^{2}) + j
. A - r . j , while in B-I NED the Lagrangian has the form :
L_{BI} = b^{-2} [(1 - Ö[1 + b^{2}(B^{2}-E^{2}) - b^{4}.(B.E)^{2}]] + j.A - r.f . | (1.119´) |
In the limit b -> 0 we get the standard
Maxwell electrodynamics, for b> 0 we are in the variant of
nonlinearity.
^{ }The equations of the electromagnetic
field in the BI model follow from the principle of the smallest
action with the Lagrangian L_{BI} (1.119´) :
(1.120) |
which are formally the same as standard linear Maxwell's equations (1.38-41). Nonlinearity is "hidden" in quantities D ("electrical induction") and H ("magnetic intensity") :
D
= ¶L/¶E = [E + b^{2}(B.E) B]/Ö[1 + b^{2}(B^{2}-E^{2}) - b^{4}.(B.E)^{2}]] , H = ¶L/¶B = [B - b^{2}(B.E) E]/Ö[1 + b^{2}(B^{2}-E^{2}) - b^{4}.(B.E)^{2}]] , |
(1.121) |
which contain factor "b". This is
analogous to the electrodynamics of the continuum,
where the relationships between E and D
= E e , B and H = B/m contain the
material coefficients of electrical permittivity e and magnetic
permeability m (including possibly inhomogeneities and nonlinearities
of polarization and magnetization).
^{ }The solution of the field equations
(1.120-121) for a spherical symmetric electrostatic field
leads to a functional dependence of the potential F and the
intensity E of the electric field on the radial distance r
from point charge Q :
(1.122) |
The integral contained in the function F(r) for the potential can be explicitly expressed using two types of special functions :
where "F(...)" is the Legender
elliptic function of the 1st kind and " _{2} F_{1} (...)" is the
so-called hypergeometric function (more
complex power series from Q^{2} b^{2} / r^{4} ). Specific
values of these complex functions can be found for physical
calculations in special tables, recently there are also computer
programs for them.
You can also write:
E(r) = q/Ö[r_{o}^{4} +
r^{4}] ; E_{max} = 1/b , where r_{o} = Ö (q.b/4p) is the " effective radius of the point charge " . F(r) = F(r/r_{o}) .q/r_{o} ( F is a combined elliptic function) , a value is based numerically F(0) = 1.854 E_{max} . |
(1.122b) |
These dependences of electric potential and intensity in the B-I NED model (dashed curves in Fig.1.8) are at greater distances r similar to the standard Coulomb's law (solid curves in Fig.1.8). However, they differ significantly at small distances: while according to Coulomb's law, with the approximation r -> 0, the potential and intensity increase to infinity, in the BI model, the final values reach F(r = 0) = F_{max} , E (r = 0) = E_{max} ~ 1 / b.
Fig.1.8. Dependence of potential F (left) and intensity E (right) of electric field on distance r from point charge (electron) according to standard Coulomb's law ( ^{_____} ) and according to nonlinear B-I electrodynamics ( ........ ). |
The specific value of the basic parameter "b" in nonlinear BI electrodynamics can be determined from the physical analysis of the electric field of an electron - the electron is a fundamental elementary particle without internal structure - is not "composed" of anything, from classical physics it can be considered a "field object". It can therefore be physically require all the rest mass of the electron m_{e} was of electrical origin. The rest energy of an electron m _{e} .c ^{2} with charge q = e should therefore be given in classical physics by the total energy of its electric field : m_{e}.c^{2} = e . _{0}n^{A}dr/Ö[r_{o}^{4} + r^{4}]. For the electron we get values :
r_{0e} = 1,2361.e^{2}/m_{e}c^{2} = 2,28 . 10^{-13} cm ; E_{max} = 1,187.10^{20} V/m . |
(1.123) |
The value of E_{max} applies not only to the electron, but in B-I
electrodynamics it is the fundamental maximum
possible value of the electric field intensity, given by the
parameter "b".
In classical physics, BI electrodynamics is considered primarily
as an electron model .
^{ }From a physical
point of view - atomic, nuclear and particle physics - the
questions of "size" or particle dimensions of the
microworld are discussed in §1.5 "Elementary particles and accelerators", passage "Size, dimensions and shape of particles? -Problematic! " in monograph "Nuclear physics and
physics ionizing radiation".
Recently, modified variants have sometimes been
introduced ^{ }nonlinear electrodynamics of NED with Lagrangian in exponential
or logarithmic form :
L_{NED} = b^{-2}[exp(-F_{ik}F^{ik}/b^{2}) - 1] or L_{NED} = -8 b^{-2} ln[(1 + F_{ik}F^{ik}/(8b^{2})], for the
purpose of wider possibilities of gravitational-electrodynamic
solutions...
^{ }Nonlinear electrodynamics, although
so far only theoretical and hypothetical,
provides interesting possibilities of solutions in theory^{ }electrically
charged black holes, where under certain circumstances
it allows to get rid of singularities (!) - §3.5, passage "Reissner-Nordstrom solution with nonlinear
electrodynamics", §3.6,
passage "Kerr-Newman geometry with nonlinear
electrodynamics" and
mention in §4. .. .... . It is some hope
that the nonlinearity of electrodynamics could "disturb"
the nonlinearity of gravity in OTR to give a more
realistic regular physical solution..?.. -
will certainly be the subject of future theoretical research. .. .
Whether this is somewhere in nature "is realized"
is not known yet - there is no hope for experimental verification
in the foreseeable future...
Classical, quantum and
gravitational nonlinear electrodynamics
In classical NED, the maximum possible value of the electric
field intensity E_{max} near the point charge was "artificially"
hypothetically postulated by introducing the
parameter "b" of nonlinearity in the field equations.
In quantum electrodynamics, the limit electric
intensity is based physically - dynamically as a
consequence of the increased production of
electron-positron pairs from a polarized vacuum in a
strong field near the point charge.
^{ }The nonlinearity of electrodynamics,
geometrically induced by the gravitational curvature
of space-time in GTR, is another matter - it will
be analyzed in §2.4
, part "Gravitational electrodynamics and optics". The "exotic" effect is the curvature
of spacetime by the energies of the electromagnetic field
- massive electromagnetic waves within geometrodynamics
(§B3 "Classical geometrodynamics. Gravity and
topology.", Geons
, Fig.B2) .
1.5. Electromagnetic field.
Maxwell's equations |
2. General theory of
relativity - physics of gravity |
Gravity, black holes and space-time physics : | ||
Gravity in physics | General theory of relativity | Geometry and topology |
Black holes | Relativistic cosmology | Unitary field theory |
Anthropic principle or cosmic God | ||
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